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In recent years, much effort has been devoted to the construction of a propermeasure of quantum non-Markovianity. However, those proposed measures are shownto be at variance with different situations. In this work, we utilize thetheory of $k$-positive maps to generalize a hierarchy of $k$-divisibility anddevelop a powerful tool, called $k$-divisibility phase diagram, which canprovide a further insight into the nature of quantum non-Markovianity. Byexploring the phase diagram with several paradigms, we can explain the originof the discrepancy between two frequently used measures and find the conditionunder which the two measures coincide with each other.

Hierarchy of Non-Markovianity and $k$-divisibility phase diagram of Quantum Processes in Open Systems