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Memory is the fundamental form of temporal complexity: when present butuncontrollable, it manifests as non-Markovian noise; conversely, ifcontrollable, memory can be a powerful resource for information processing.Memory effects arise from/are transmitted via interactions between a system andits environment; as such, they can be either classical or quantum in nature.From a practical standpoint, quantum processes with classical memory promisenear-term applicability: they are more powerful than their memorylesscounterpart, yet at the same time can be controlled over significant timeframeswithout being spoiled by decoherence. However, despite practical andfoundational value, apart from simple two-time scenarios, the distinctionbetween quantum and classical memory remains unexplored. We first analysevarious physically-motivated candidates regarding a suitable definition forclassical memory that lead to remarkably distinct phenomena in the multi-timesetting. Subsequently, we systematically characterise the hierarchy ofmulti-time memory effects in quantum mechanics, many levels of which collapsein the two-time setting, thereby making our results genuinely multi-timephenomena.

Characterising the Hierarchy of Multi-time Quantum Processes with Classical Memory