  Entanglement-breaking channels (equivalently, measure-and-prepare channels)
are an important class of quantum operations noted for their ability to destroy
multipartite spatial quantum correlations. Inspired by this property, they have
also been employed in defining notions of "classical memory", under the
assumption that such channels effectively act as a classical resource. We show
that, in a single-system multi-time scenario, entanglement-breaking channels
are still a quantum memory resource: a qudit going through an
entanglement-breaking channel cannot be simulated by a classical system of same
dimension. We provide explicit examples of memory-based output generation tasks
where entanglement-breaking channels outperform classical memories of the same
size. Our results imply that entanglement-breaking channels cannot be generally
employed to characterize classical memory effects in temporal scenarios without
additional assumptions.
