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We provide a complete set of game-theoretic conditions equivalent to theexistence of a transformation from one quantum channel into another one, bymeans of classically correlated pre/post processing maps only. Such conditionsnaturally induce tests to certify that a quantum memory is capable of storingquantum information, as opposed to memories that can be simulated bymeasurement and state preparation (corresponding to entanglement-breakingchannels). These results are formulated as a resource theory of genuine quantummemories (correlated in time), mirroring the resource theory of entanglement inquantum states (correlated spatially). As the set of conditions is complete,the corresponding tests are faithful, in the sense that any nonentanglement-breaking channel can be certified. Moreover, they only require theassumption of trusted inputs, known to be unavoidable for quantum channelverification. As such, the tests we propose are intrinsically different fromthe usual process tomography, for which the probes of both the input and theoutput of the channel must be trusted. An explicit construction is provided andshown to be experimentally realizable, even in the presence of arbitrarilystrong losses in the memory or detectors.

A resource theory of quantum memories and their faithful verification with minimal assumptions