To characterise a physical system, we must observe it. From these observations, we construct models and estimate the parameters governing its behaviour. The difficulty of this task is greatest not in regimes of order or chaos - both of which are relatively simple to simulate - but in the intermediate regime of complexity. In temporal processes, such complexity is closely tied to the presence of memory: correlations between past and future that must be retained by any predictive model.

In this talk, I will explore two complementary roles of memory in this context. First, I will show how quantum mechanics enables more memory-efficient models of classical stochastic processes, allowing us to simulate complex dynamics with reduced resources compared to classical approaches. Focusing on the quantum resources that underlie this advantage we single out the role of quantum coherence. I will then turn to continuously monitored quantum systems, where memory plays a central role in learning: I will discuss methods for estimating system parameters from quantum jump trajectories, where past measurement outcomes influence future evolution.

Together, these results highlight a unifying perspective: in quantum information processing, memory is both a fundamental constraint and a powerful resource.

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