  In recent years, much effort has been devoted to the construction of a proper
measure of quantum non-Markovianity. However, those proposed measures are shown
to be at variance with different situations. In this work, we utilize the
theory of $k$-positive maps to generalize a hierarchy of $k$-divisibility and
develop a powerful tool, called $k$-divisibility phase diagram, which can
provide a further insight into the nature of quantum non-Markovianity. By
exploring the phase diagram with several paradigms, we can explain the origin
of the discrepancy between two frequently used measures and find the condition
under which the two measures coincide with each other.
