  Memory is the fundamental form of temporal complexity: when present but
uncontrollable, it manifests as non-Markovian noise; conversely, if
controllable, memory can be a powerful resource for information processing.
Memory effects arise from/are transmitted via interactions between a system and
its environment; as such, they can be either classical or quantum. From a
practical standpoint, quantum processes with classical memory promise near-term
applicability: they are more powerful than their memoryless counterpart, yet at
the same time can be controlled over significant timeframes without being
spoiled by decoherence. However, despite practical and foundational value,
apart from simple two-time scenarios, the distinction between quantum and
classical memory remains unexplored. Here, we analyse multi-time quantum
processes with memory mechanisms that transmit only classical information
forward in time. Complementing this analysis, we also study two related -- but
simpler to characterise -- sets of processes that could also be considered to
have classical memory from a structural perspective, and demonstrate that these
lead to remarkably distinct phenomena in the multi-time setting. Subsequently,
we systematically stratify the full hierarchy of memory effects in quantum
mechanics, many levels of which collapse in the two-time setting, making our
results genuinely multi-time phenomena.
