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Thermodynamics is a highly successful framework to describe many-particle equilibrium systems through a small number of collective variables. Yet, in quantum systems, it is still unclear how thermodynamic behaviour emerges microscopically from far-from-equilibrium initial conditions, as the evolution of closed quantum systems is unitary. While it has been realised that the distribution of correlations among many constituents is a key driving mechanism of equilibration, microscopically tracking the dynamical build-up of this process is a formidable challenge: At long times, close to equilibrium, hydrodynamics allow for an efficient description in terms of few collective degrees of freedom, but intermediate time scales are characterised by high complexity and pose serious challenges even for the most advanced theoretical and computational techniques. Significant progress in understanding the microscopic evolution of complexity and the emergence of equilibration can only be driven by a strong confluence of theoretical and experimental endeavours.

An exciting inroad in this context is opened by a new generation of quantum simulation machines, where many-body systems can be engineered at the level of individual constituents. Amongst the possible implementations, quantum simulators based on ultracold atoms offer an unprecedented magnifying glass for probing coherent out-of-equilibrium dynamics over long time scales.

In CoQuS, we will follow a bottom-up approach and implement atomic simulators with a small number of precisely assembled constituents. Such small-scale quantum simulators are excellent playgrounds to shed light on the reciprocity between equilibration and complexity, both from the theoretical and the experimental point of view, as they allow for tracing an explicit connection between microscopic dynamics and non-equilibrium statistical mechanics. To this aim, we will leverage some of the most powerful tools in statistical and computational physics: (i) fluctuation-dissipation relations and (ii) complexity theory. A new experimental platform based on ultracold two-electron atoms will enable novel diagnostics for quantifying equilibration and complexity, owing to an increased coherent control and ultra-precise probing capabilities. Experimental explorations will be guided by state-of-the-art theoretical approaches, allowing us also to devise new strategies and establish presently lacking, rigorous connections between complexity and non-equilibrium dynamics.