Disadvantages of this crypto system Consider the following


Alice and Bob agree to an encyption system based on a solution to a complex chemical simulation
Alice announces publicly that her "key" is the chemical tryptophan. Therefore any solutions with tryptophan as an input belong to Alice.
A public distributed ledger exists.
Alice publishes her tryptophan choice on the ledger.
Now alice can create simulations combining tryptophan with other unclaimed chemicals. 
The simulations are computationally expensive and can't be readily solved by quantum attacks. (Nobody is proposing a cheap way to simulate chemical reactions using quantum computers, and replicating the reactions directly requires manual effort). 
Alice does not publicly publish the solution. She only publicly publishes that she has a solution for converting tryptophan to another specific chemical.
Bob values that solution and offers to trade with Alice.
Alice privately transfers the solution to Bob as payment for a good/service
If Alice decides to double-spend the solution, Bob can publicly publish it (and make it worthless).
If Eve steals the solution either can also reveal it and make it worthless.
Therefore as long as it is kept secret, the solution to this chemical reaction serves as a transferable store of value that is not subject to attack.
The value of the solution depreciates as it is spent more because more people have it. 


What are the problems with this?

Also, you could also have it so the data in the solution is so enormous that it cannot be easily published or stolen. Bob would receive permission to access Alice's cluster to use the data but it would be impractical to steal it.
A chemical simulation based proof of work? Interesting, but let's ignore how the mathematics work for generating a solution based on my input key being "rhubarb". And zero knowledge based verification of said vegetable? Seems tricky. Must we assume that a 'simulation' is akin to a one way function/hash/mining?  

But I would focus on #8, valuation.  A floating currency is set by the market environment, not individual consumers on a per contract basis.  How can Bob set a realistic price for your ChemiCoin(TM)? Especially as they're not underwritten by tangible assets.  I think that I'd mow your lawn for 100 ChemiCoin(TM). But if no one else values them similarly, I've effectively consumed my resources for no reward.  That's bad capitalism and realisation would be neigh on impossible.

#8, part 2. How can Alice validate execution of the contract if the other party is not reciprocating with ChemiCoin(TM) transactions on the ledger?  Did I really mow your lawn?

And on #13.  What use treasure if it can't be spent? At least the Pharaohs could spend their accumulated gold and jewels in the Underworld.  The more ChemiCoin(TM) I possess, the more I want to spend them. But they depreciate accordingly.  So why amass them in the first place?  I may be getting poorer the richer I get, and don't mean  morally.

It all seems very tricky mathematically and economically, with a highly divisive set of cryptographic ideas.