We propose an information theoretically secure secret-key quantum money scheme in which the verification of a coin is classical and consists of only one round; namely, a classical query from the user to the bank and an accept/reject answer from the bank to the user. A coin can be verified polynomially (on the number of its qubits) many times before it expires. Our scheme is an improvement on Gavinsky’s scheme, where three rounds of interaction are needed and is based on the notion of quantum retrieval games. Moreover, we propose a public-key quantum money scheme which uses one-time memories as a building block and is computationally secure in the random oracle model. This construction is derived naturally from our secret-key scheme using the fact that one-time memories are a special case of quantum retrieval games.