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Although quantum coherence is a basic trait of quantum mechanics, thepresence of coherences in the quantum description of a certain phenomenon doesnot rule out the possibility to give an alternative description of the samephenomenon in purely classical terms. Here, we give definite criteria todetermine when and to what extent quantum coherence is equivalent tonon-classicality. We prove that a Markovian multi-time statistics obtained fromrepeated measurements of a non-degenerate observable cannot be traced back to aclassical statistics if and only if the dynamics is able to generate coherencesand to subsequently turn them into populations. Furthermore, we show withsimple examples that such connection between quantum coherence andnon-classicality is generally absent if the statistics is non-Markovian.

Coherence and non-classicality of quantum Markov processes