Why does the prover in an interactive proof run in exponential time with the space of the computation? I'm currently trying to gain an understanding of verifiable computation and the strengths and weaknesses of past approaches. 

In particular, I've been watching the following YouTube video which talks about interactive proofs as a way of verifying knowledge of a computation. On the slide appearing during time 11:55, it mentions that the prover requires exponential time in the space of the computation. Could anyone explain how this is the case? 

I'm also confused as to the statement which says that every deterministic computation has an interactive proof.  

I've tried reading the papers below but can't seem to find a clear explanation of these results. Any help would be much appreciated!

References:


  Lund, Carsten, et al. "Algebraic methods for interactive proof
  systems." Journal of the ACM (JACM) 39.4 (1992): 859-868.
  
  Shamir, Adi. "Ip= pspace." Journal of the ACM (JACM) 39.4 (1992):
  869-877.
The whole point of verifiable computation is to allow the verifier to verify that the computation was performed correctly without performing the computation himself. Since as usual we associate "polynomial-time" with "efficiently computable", if the computation takes poly time there is no point in using verifiable computation, since the verifier can just as well perform the computation himself.

In proof systems in general, if the verifier can determine by himself the truth of the statement in question, there is no need for a prover at all.