Referring to Farhi, Gosset, Hassidim, Lutomirski, and Shor's "Quantum Money from Knots," a mint $\mathcal{M}$ generates a run of coins, including, say, $(s,|\$\rangle)$, using a quantum computer to mint $|\$\rangle$ while publishing the public serial number $s$. A merchant $\mathcal{V}$ is able to verify that $|\$\rangle$ corresponds to $s$, and is a valid money state.
The merchant $\mathcal{V}$ is required to perform quantum operations on $|\$\rangle$ to make sure that $|\$\rangle$ corresponds to $s$ and is a valid money state.
But if the merchant $\mathcal{V}$ is capable of performing the verification, then would she not have all quantum capability to mint her own coin in the first place?
The barrier to entry to minting quantum coins does not seem that much different to verifying quantum coins. Thus, we have a situation wherein anyone with a quantum computer capable of verifying such quantum coins can mint their own coins.
If so, a question becomes, how would the market determine the value of a quantum coin, potentially from different merchants or minters? Would the "oldest" quantum coin be valued more than newer coins? Or would a coin with an interesting serial number $s$ be valued more? Or the coin used in some famous transaction?
I would imagine a number of publicly available lists of serial numbers, one for each mint/verifier. If I have a quantum computer, I would be motivated to mint my own coin and publish the serial number. The market can decide that "this coin from this mint is worth more than that coin from that mint," but how would the market decide? It seems interesting.