The problem of memory checking considers storing files on an unreliable public server whose memory can be modified by a malicious party. The main task is to design an online memory checker with the capability to verify that the information on the ser
ver has not been corrupted. To store n bits of public information, the memory checker has s private reliable bits for verification purpose; while to retrieve each bit of public information the checker communicates t bits with the public memory. Earlier work showed that, for classical memory checkers, the lower bound s*t in Omega(n) holds. In this article we study quantum memory checkers that have s private qubits and that are allowed to quantum query the public memory using t qubits. We prove an exponential improvement over the classical setting by showing the existence of a quantum checker that, using quantum fingerprints, requires only s in O(log n) qubits of local memory and t in O(polylog n) qubits of communication with the public memory.
