  Non-Markovian processes may arise in physics due to memory effects of
environmental degrees of freedom. For quantum non-Markovianity, it is an
ongoing debate to clarify whether such memory effects have a verifiable quantum
origin, or whether they might equally be modeled by a classical memory. In this
contribution, we propose a criterion to test locally for a truly quantum
memory. The approach is agnostic with respect to the environment, as it solely
depends on the local dynamics of the system of interest. Experimental
realizations are particularly easy, as only single-time measurements on the
system itself have to be performed. We study memory in a variety of physically
motivated examples, both for a time-discrete case, and for time-continuous
dynamics. For the latter, we are able to provide an interesting class of
non-Markovian master equations with classical memory that allows for a
physically measurable quantum trajectory representation.
