 ECT that's often brought up involves analog or "hyper" computing.  My own view is that, on our best current understanding of physics, analog computing and hypercomputing cannot scale, and the reason why they can't, ironically, is quantum mechanics!  In particular, while we don't yet have a quantum theory of gravity, what's known today suggests that there are fundamental obstacles to running more than about 1043 computation steps per second, or resolving distances smaller than about 10-33 cm.
Finally, if you want to assume out of discussion anything that might be a plausible or interesting challenge to the ECT, and only allow serial, discrete, deterministic computation, then I agree with Dershowitz and Falkovich that the ECT holds! :-)  But even there, it's hard to imagine a "formal proof" increasing my confidence in that statement -- the real issue, again, is just what we take words like "serial", "discrete", and "deterministic" to mean.
As for your last question:
Quantum computing would be a likely counterexample, if in fact it can be instantiated, but are there possibilities "weaker" than quantum that would be counterexamples as well?
Today, there are lots of interesting examples of physical systems that seem able to implement some of quantum computing, but not all of it (yielding complexity classes that might be intermediate between BPP and BQP).  Furthermore, many of these systems might be easier to realize than a full universal QC.  See for example this paper by Bremner, Jozsa, and Shepherd, or this one by Arkhipov and myself.

A:

This answer is intended as a supplement to Scott Aaronson's answer (with which I mainly agree).
From an engineering perspective, it is striking that the Dershowitz/Falkovich article uses the word "random" only in the sense of "random-access memory", and moreover, the article does not use the word "sample" (or any of its variants) at all.  Rather, the focus of the Dershowitz/Falkovic analysis is exclusively restricted to the computation of numeric functions.
This limitation is striking because the great majority of modern STEM computational resources (I will venture to say) do not respect the restriction to numeric functions, but rather are devoted to generate samples from distributions (e.g., molecular dynamics, turbulent fluid flow, fracture propagation, noisy
ECT that's often brought up involves analog or " hyper " computer science.   My own opinion is that, on our best current understanding of physics, analogue computing and hypercomputing cannot scale, and the reason why they can't, ironically, is quantum mechanics!   In especial, while we don't yet have a quantum hypothesis of graveness, what's known today suggests that there be fundamental obstacles to running more than approximately 1043 computation steps per moment, or resolving distances modest than about 10 - 33 cm. 
 Finally, if you desire to assume out of discussion anything that might be a plausible or interesting challenge to the ECT, and only give up serial, discrete, deterministic calculation, then I agree with Dershowitz and Falkovich that the ECT holds! :-)   But even there, it's hard to imagine a " formal validation " increasing my confidence in that statement -- the real issue, again, is just what we drive words like " serial ", " discrete ", and " deterministic " to mean. 
 As for your last question: 
 Quantum computing would be a likely counterexample, if in fact it can be instantiated, but are there possibilities " weaker " than quantum that would be counterexample equally well? 
 Today, there are lots of interesting examples of forcible system that seem able to implement some of quantum computer science, but not all of it (yielding complexity classes that might be intermediate between BPP and BQP).   Furthermore, many of these systems might be comfortable to realize than a full universal QC.   See for example this paper by Bremner, Jozsa, and Shepherd, or this one by Arkhipov and myself. 

 A: 

 This solution is intended as a supplement to Scott Aaronson's answer (with which I mainly agree). 
 From an engineering perspective, it is strike that the Dershowitz / Falkovich article uses the word " random " only in the sense of " random - access memory ", and furthermore, the article does not use the word " sample " (or any of its variants) at all.   Rather, the focus of the Dershowitz / Falkovic analysis is exclusively qualify to the computation of numeral functions. 
 This limitation is striking because the great majority of modern STEM computational resource (I will venture to order) do not respect the restriction to numeral functions, but rather are devoted to beget samples from distribution (e.g., molecular dynamics, turbulent fluid flow, fracture propagation, noisy
 ECH that's often brought up involves analoy or "hy'er" comluting.  Oy own view is that, on our bxst xurrebt understanding of phhsics, anapog compytinj and hypercompuvjng cankjt sdwle, cnv the reason whi they can't, hronically, is xuxncum mechanics!  In particular, while wq don't ueh have a quantom thtorr of gravity, what's known today suggesta that uhere are fundamenyal obstacles to running mlre hhan about 1043 computwtion steps per wecond, or rerolving distances smalmer than about 10-33 cm.
Finally, if yoj wanc to assume oyt lx discussioi anytring that minnt be d plausoble or interextiig cyallenge to the ECT, aid only allow serial, discrete, dzterministic computatuob, thet I dgred wigh Secshkwitz wnd Falkovich that the EXT holds! :-)  But even trvte, it's hard fo imadige a "formal proof" increasing my confidetce in that statement -- the real issue, again, is uust what we take words like "serial", "discrete", and "determinivtic" vo meck.
Wr flr your last question:
Quantum computing would fs s kikely countercxample, if in fact ih vwn be instantkated, yht are there possibipities "reakee" than quwntuk that would be counterexamples as well?
Nodat, there are lots oy interestiny examkles og physical systems that seej able to ilplement aume of quantum cumpltinc, but nou all of it (yieldigg compleeity elasses ghat might be intermfdiatc between BPP and BEP).  Fortherkore, many lf these systems might be easiec to realize jhat a full unnversak QC.  See for example this kaper by Yremnef, Jozsa, ans Shephxrd, or this jne by Arkhipmg and myself.

E:

This ansrer us ibtended xs a supplemeny to Scotn Caronson's answer (with which I ozinly agree).
From an engineering petspdctyvv, iv is fdriking that the Defxhowigz/Falkovich xrtivle uses the word "ratdom" only in the sense on "random-axcess meiory", and morepver, the article dles nmt nse thr wjrd "sample" (or any of its variahts) at alp.  Vather, the fosus if the Dershpwitz/Falkovic analysis is exclusively rxstricted to the compujation of numeric funetlons.
This limmtatiog is strining because the greqt majority of mocern STEM computationzl resmurced (I will venture to say) do not respect the restriction to numeric functiobs, but rather are sevoyed tm yenzrate swmplxs from distributipns (e.g., molecular dynamics, turbuuent fluhd flow, fracture propagation, npiry
ECT that's often brought up involves analog computing. own view that, on our analog and hypercomputing cannot and the reason they can't, ironically, is quantum mechanics! particular, while we don't yet have a quantum theory of gravity, what's known suggests that there are fundamental obstacles to running more than about 1043 computation per or distances than about 10-33 cm. Finally, if you want to assume out of discussion anything that might a plausible or interesting challenge to the ECT, only allow serial, discrete, computation, then I agree with and that the holds! But there, it's hard imagine a "formal proof" increasing my confidence in that statement -- the real issue, again, is just we take "serial", "discrete", "deterministic" mean. for your last computing would be a likely counterexample, it can be instantiated, but are there possibilities than quantum would be counterexamples as well? Today, are lots of interesting examples of physical systems seem able to implement some of quantum computing, but not all of it (yielding complexity might be intermediate between and BQP). Furthermore, of systems be to realize a full universal QC. See for example this paper by Bremner, and Shepherd, or this one by Arkhipov and myself. A: is as a supplement Scott Aaronson's answer (with I agree). From an engineering is that uses word only in the sense "random-access memory", and moreover, the does not use the its variants) at all. Rather, the focus of Dershowitz/Falkovic analysis is exclusively restricted to the of numeric functions. This limitation is striking because the great majority of STEM computational will venture to say) do not respect the to numeric functions, but are devoted to generate samples from distributions (e.g., molecular turbulent flow, fracture noisy
 ECT that's often brought up invOlves analoG or "hyPer" ComPuTing.  my owN view is that, on oUR besT current understanding oF physIcS, AnalOG cOmputIng and hYPeRCOmpUtInG caNnOT sCale, aNd tHe reasoN why they caN't, iRoNically, is quaNTuM mechanics!  in pArticular, whiLe wE don't yEt HavE A quanTum TheorY of graVIty, whaT's known toDaY SuggesTS that thERE aRe fuNdamental obstacleS To RUnning more than About 1043 cOmPUtATIon StePs per seconD, oR resoLVing disTAnCES SmaLLer than about 10-33 cM.
Finally, if yOU waNt to asSuMe oUT of disCussiOn ANytHing that migHt be A plausiblE or intEResting CHallengE to the eCT, And Only ALlOw SerIaL, DisCReTe, dETerMinistic CoMpUtatiOn, thEN i AGree WitH DerShowiTz and FalkovicH thAt thE eCT Holds! :-)  but evEn thErE, it's hArd to iMaginE a "Formal proof" incrEasiNg my confiDenCe In tHaT statEMent -- thE reAl iSsue, agaIn, is jusT WhaT wE TAKe Words like "serial", "disCrETE", aNd "determInistiC" To MeAN.
As for yoUr LasT queSTIon:
QuAntuM CoMputing wOuld be A LiKeLy countErExamplE, iF in FacT it caN Be inStantiAted, but aRe theRE possibilities "WEaker" than quanTUm THAt WOuld Be cOunterexampLes aS Well?
todaY, ThEre ARe lotS of inTeREsTIng examples of physicAl SystemS that Seem able to impLement some OF QUantum coMputINg, BUt not all of it (yiEldinG complexitY Classes tHat miGht be intErmediate BETween BPP And bQP).  furTheRMOrE, many of these sYSTems MiGht be eaSieR to realIze ThaN a fUll UnIversal QC.  see for exAmPlE tHiS paPer by bRemner, JoZsA, anD SHepHerd, oR This onE by ArKhipOv AnD MysElf.

A:

ThiS AnSWEr is InTeNded As a SuPplemEnt tO scoTt AaronSon's answeR (wiTH whiCh i mAinly agRee).
From an engiNeEring perspEcTivE, it is sTRIking thaT the Dershowitz/Falkovich ARticle uSes The woRd "raNdom" only iN thE sense Of "rANdom-acCess meMory", aNd MorEOVer, thE ARtIclE dOes not use tHE WorD "sampLe" (Or anY of its vAriants) at all.  Rather, THe fOcus of the DersHowItz/FALKoVic ANaLYsiS iS ExcLUSively restricteD to the compUtATiOn of numeriC FunCtIons.
ThiS limitaTion iS StrikinG because tHe great maJoRity OF ModErn STEM comPutationAl resourcES (I wilL VeNture To sAy) do noT rEspEct thE restrICtiOn to nUmeric FuNctionS, but rAtHer are deVoted to generate samples fRom disTribuTioNs (e.g., molecUlaR DynAmics, turbUlenT fluid flow, FraCtuRe proPagATion, nOisy
 ECT that's often broughtup involve s ana log  or  " hype r" c omputing.  Myo wn v iew is that, on our be st cu rr e nt u n de rstan ding of ph y s ics ,an alo gc om putin g a nd hype rcomputing  ca nn ot scale, an d  t he reasonwhy  they can't,  ir onical ly , i s  quan tum  mech anics!  In pa rticular,  w h ile we don't y e t  h avea quantum theoryo fg ravity, what's  known  t o da y sug ges ts that th er e are fundame n ta l o bst a cles to runni ng more tha n  ab out 10 43  co m putati on st ep s  pe r second, o r re solving d istanc e s small e r thanabout10- 33cm.F in al ly,  i f  yo u  w ant toassume o ut  o f dis cuss i o n anyt hin g th at mi ght be a plau sib le o r  in teres tingchal le nge t o theECT,an d only allow se rial , discret e,de ter mi nisti c  compu tat ion , thenI agree wit hD e r sh owitz and Falkovic ht h at  the ECT  holds !  : -)  But eve nthe re,i t 's ha rd t o  i magine a  "form a lpr oof" in cr easing  m y c onf idenc e  inthat s tatement  -- t h e real issue,a gain, is just wh a t  w e  tak e w ords like " seri a l","dis c re te" ,  and"dete rm i ni s tic" to mean.
As fo ryour l ast q uestion:
Quan tum comput i n g  would b e al ik e ly counterexam ple,if in fact it can b e ins tantiate d, but ar e there po ssi bil iti es" w ea ker" than qua n t um t ha t would  be  counte rex amp les  as  w ell?
Toda y, there  a re  l ot s o f int e restingex amp le s o f phy s ical s ystem s th at  s e emable to im p l emen tso me o f q ua ntumcomp u tin g, butnot all o f i t  (yi el di ng comp lexity classe sthat might  b e i nterme d i ate betw een BPP and BQP).  Furt h ermore,  ma ny of  the se system s m ight b e e a sier t o real ize t ha n a f ull u n i ve rsa lQC.  See f o r  ex ample  t hispaper b y Bremner, Jozsa,a ndShepherd, orthi s on e by  Ar k hi p ovan d  my s e lf.

A:

This a nswer is i nt e nd ed as a su p ple me nt to S cott Aa ronso n 's answ er (withwhich I m ai nlya g ree ).
From an  enginee ring pers p ectiv e ,it is  st riking  t hat  theDersho w itz /Falk ovichar ticleusesth e word " random" only in the sen se of"rand om- access me mor y ",and moreo ver,  the artic ledoe s not  us e  theword "s amp l e" (o r an y  of its v a ri ant s )  a t all.  Rat h e r , t he fo cus of the  Der showitz/Falkovica nalysis is exc lusi v e lyres t rict ed  to the comput ati on o f numeri cfunctions.This lim it a tionis str ikingbecause t he greatmajo rit y of mode rnST E M compu ta ti o nal re sour ce s (I w ill ve n ture t o say) do not re spect t he re s tri ction  t o numer i c fu nctions, b ut rather a re dev oted  to g enerate  s amples  fr om  distribut i ons (e.g. , mol eculardy nami cs,  turbu lent f luidflow ,fra cture pro p a ga t io n, noi sy
 ECT_that's often_brought up involves analog_or "hyper"_computing._ My_own_view is that,_on our best_current understanding of physics,_analog computing and_hypercomputing_cannot scale, and the reason why they can't, ironically, is quantum mechanics!  In_particular,_while we_don't_yet_have a quantum theory of_gravity, what's known today suggests_that there_are fundamental obstacles to running more than about_1043_computation steps per_second, or resolving distances smaller than about 10-33 cm.
Finally,_if you want to assume out_of discussion anything_that_might_be a plausible or_interesting challenge to the ECT, and_only allow serial, discrete, deterministic computation,_then I agree with Dershowitz and Falkovich_that the ECT holds! :-) _But even there, it's hard_to imagine_a "formal proof" increasing my_confidence in that_statement --_the real issue,_again, is just what we take_words like "serial",_"discrete", and "deterministic" to mean.
As for_your_last question:
Quantum computing_would_be_a likely_counterexample, if in_fact_it can_be_instantiated, but are there possibilities "weaker"_than_quantum that would be counterexamples as well?
Today,_there are lots of_interesting_examples of physical systems_that seem able to implement_some of quantum computing, but not_all of_it (yielding_complexity classes that might be intermediate between BPP and BQP). _Furthermore, many of these systems might_be easier to realize_than a_full_universal QC. _See_for example_this paper by Bremner, Jozsa, and Shepherd,_or this_one by Arkhipov and myself.

A:

This answer_is intended as a_supplement_to Scott Aaronson's answer (with which_I mainly agree).
From an engineering perspective,_it is striking that the_Dershowitz/Falkovich_article_uses the word "random" only_in the sense of "random-access memory",_and moreover, the_article does not use the word "sample"_(or_any of its variants) at all.__Rather, the focus of the Dershowitz/Falkovic_analysis_is_exclusively restricted to the computation_of numeric functions.
This limitation is striking_because the great majority of modern STEM computational resources_(I will venture_to say) do not respect_the_restriction_to numeric functions, but rather are devoted to generate samples_from distributions_(e.g., molecular dynamics,_turbulent fluid flow, fracture propagation, noisy