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 non-
Markovian 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.