When Is a Non-Markovian Quantum Process Classical?

Quantum mechanics has been developed almost a century ago, due to the need of describing experimental results that were not explicable with classical physics. This lead to a surge of theoretical research aimed at reconciling our intuition, based on experiences of our everyday life, and the new theory. One main result of that research is that some seemingly intuitive assumptions are not reconcilable with the predictions of quantum mechanics and the subsequent experimental tests showed that it is quantum mechanics that prevails. It is this new confidence in the theory which is now leading to new research that aims at making use of these special traits, for building better sensors, quantum simulators and computers. By now, it is known that entanglement, discord and coherence play a central role in the quantum advantage. However, how does it work, exactly and how much is the advantage that we get due to the quantum properties?
Using the defining properties of stochastic processes, we were recently able to make precise how it is coherence and discord that make it impossible to simulate some quantum processes by classical ones. Using these insights, one gets meaningful quantifiers of how well the best possible classical model describes a given quantum experiment.