When Is a Non-Markovian Quantum Process Classical?

More than a century after the inception of quantum theory, the question of which traits and phenomena are fundamentally quantum remains under debate. Here, we give an answer to this question for temporal processes that are probed sequentially by means of projective measurements of the same observable. Defining classical processes as those that can, in principle, be simulated by means of classical resources only, we fully characterize the set of such processes. Based on this characterization, we show that for nonMarkovian processes (i.e., processes with memory), the absence of coherence does not guarantee the classicality of observed phenomena; furthermore, we derive an experimentally and computationally accessible measure for nonclassicality in the presence of memory. We then provide a direct connection between classicality and the vanishing of quantum discord between the evolving system and its environment. Finally, we demonstrate that-in contrast to the memoryless setting-in the non-Markovian case, there exist processes that are genuinely quantum; i.e., they display nonclassical statistics independent of the measurement scheme that is employed to probe them.