  We provide a complete set of game-theoretic conditions equivalent to the
existence of a transformation from one quantum channel into another one, by
means of classically correlated pre/post processing maps only. Such conditions
naturally induce tests to certify that a quantum memory is capable of storing
quantum information, as opposed to memories that can be simulated by
measurement and state preparation (corresponding to entanglement-breaking
channels). These results are formulated as a resource theory of genuine quantum
memories (correlated in time), mirroring the resource theory of entanglement in
quantum states (correlated spatially). As the set of conditions is complete,
the corresponding tests are faithful, in the sense that any non
entanglement-breaking channel can be certified. Moreover, they only require the
assumption of trusted inputs, known to be unavoidable for quantum channel
verification. As such, the tests we propose are intrinsically different from
the usual process tomography, for which the probes of both the input and the
output of the channel must be trusted. An explicit construction is provided and
shown to be experimentally realizable, even in the presence of arbitrarily
strong losses in the memory or detectors.
