As with quantum computing itself, storing quantum money would require low enough error rates that you could successfully apply quantum error-correction (if error rates are too high, you introduce more errors during error-correction than you can correct). Before you hit this point you can’t do the computations anyway, and after this point it is only a tiny bit harder to keep qubits around indefinitely. (Modern QKD works because you don’t have to do any computations on the qubits, in addition to not having to store them.)
As with quantum computing itself, storing quantum money would require low enough error rates that you could successfully apply quantum error-correction (if error rates are too high, you introduce more errors during error-correction than you can correct). Before you hit this point you can’t do the computations anyway, and after this point it is only a tiny bit harder to keep qubits around indefinitely. (Modern QKD works because you don’t have to do any computations on the qubits, in addition to not having to store them.)