Unified trajectory criterion for quantum and classical non-Markovianity

We introduce a simple and universal criterion for identifying non-Markovian dynamics in open quantum systems. Our approach is based on the geometry of the system's trajectory set, i.e. the collection of all possible dynamical trajectories generated by a master equation. We show that a quantum master equation is non-Markovian if and only if its trajectory set contains a self-intersecting trajectory. As self-intersection is invariant under time reversal, this criterion implies that Markovianity itself is a time-reversal–invariant property, unlike many existing criteria based on information flow or complete positivity. By analyzing the structure of trajectory sets, we classify dynamics into three distinct types: strictly Markovian, initial-state Markovian, and non-Markovian. Through multiple examples, we compare our criterion with existing ones and show how they fail to capture Markovianity and non-Markovianity in various cases.