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.)