  We analyze recent approaches to quantum Markovianity and how they relate to
the proper definition of quantum memory. We point out that the well-known
criterion of information backflow may not correctly report character of the
memory falsely signaling its quantumness. Therefore, as a complement to the
well-known criteria, we propose several concepts of elementary dynamical maps.
Maps of this type do not increase distinguishability of states which are
indistinguishable by von Neumann measurements in a given basis. Those notions
and convexity allows us to define general classes of processes without quantum
memory in a weak and strong sense. Finally, we provide a practical
characterization of the most intuitive class in terms of the new concept of
witness of quantum information backflow.
