Temporal Quantum Correlations In dynamical non-equilibrium situations there is not just entanglement that plays a fundamental role. As noted already by Heisenberg, measurements of “complementary” observables are incompatible, which means that one necessarily disturbs the other if performed in a sequence. In this case it is perhaps even more counter-intuitive the fact that this incompatibility can be also seen as a resource for generating temporal sequences of outputs, like bit strings. This resource can be quantified in terms of the minimal number of distinguishable states available to a machine, which is a notion of memory. Furthermore, there is a distinction if these states can be manipulated via a classical finite-state machine or a quantum one, which calls for the question: what is the classical memory-cost of simulating a quantum finite-state machine? This can be quantified by looking at nontrivial bounds for the probability of generating complex sequences. In this framework, we look also for connections between different quantifications of temporal correlations, like non-Markovianity or error-disturbance trade-offs, which also relate to foundational questions such as the existence of extensions of so-called Macro-realist models for simulating quantum temporal sequences.