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I will go over our recent research, which bridges quantum many-body physics and quantum simulation through two main pillars. The first focuses on investigating exotic far-from-equilibrium phenomena in lattice gauge theories, uncovering properties of dynamical phase transitions and criticality. The second aims to develop experimentally feasible schemes to probe such dynamics on state-of-the-art quantum hardware, including cold atoms and molecules, superconducting qubits, and trapped ions. We will go over a series of experiments highlighting the interplay between these two pillars.