Similar to Isaac's unlisted question, but with slightly different criteria.
Such a cryptocurrency would need to include the following features:
The ability to spend a coin depends only on possession of some quantum "token".
Tokens must be publically verifiable
Anyone with a fault-tolerant quantum computer and a token can transfer the token to anyone with another quantum computer, as long as they can transfer quantum data
Starting with a certain amount of money, it is infeasible to double-spend or create more money.
Finalization of transactions in any location takes less than 10 minutes.
Based only on well-accepted post-quantum cryptographic assumptions at the time the design is released (recursive SNARKs and lattice-crypto are examples of things currently ok as long as they don't get broken, money-from-knots is not currently ok, but could become ok with stronger justification)
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@ArunJohnson No. It's just that the transfer between parties has to take less than 10 minutes, and it can't be possible for double-spends to take place.
@Ansel I sort of view the scalability as built-in - if a person can transfer a token without any external internet access, then the system by default scales to however many users it needs to.