When is a non-Markovian quantum process classical?

More than 90 years after the inception of quantum theory, the question of which traits and phenomena are fundamentally quantum remains under debate. Here, we comprehensively answer this question for temporal processes. 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 non-Markovian processes (i.e., processes with memory), the absence of coherence does not guarantee the classicality of observed phenomena. We then provide a direct connection between classicality and the vanishing of quantum discord between the evolving system and its environment, thus establishing a clear link between discord---a trait that is considered to be exclusively quantum---and the non-classicality of observable phenomena in temporal processes.