Significance Quantum phase transitions are characterized by a dramatic change of the ground-state behavior; famous examples include the appearance of magnetic order or superconductivity as a function of doping in cuprates. In this work, we explore how a system dynamically crosses such a transition and in particular, investigate in detail how coherence emerges when an initially incoherent Mott insulating system enters the superfluid regime. We present results from an experimental study using ultracold atoms in an optical lattice as well as numerical simulations and find a rich behavior beyond the scope of any existing theory. This quantum simulation of a complex many-body system is an important stepping stone for a deeper understanding of the intricate dynamics of quantum phase transitions. The dynamics of quantum phase transitions pose one of the most challenging problems in modern many-body physics. Here, we study a prototypical example in a clean and well-controlled ultracold atom setup by observing the emergence of coherence when crossing the Mott insulator to superfluid quantum phase transition. In the 1D Bose–Hubbard model, we find perfect agreement between experimental observations and numerical simulations for the resulting coherence length. We, thereby, perform a largely certified analog quantum simulation of this strongly correlated system reaching beyond the regime of free quasiparticles. Experimentally, we additionally explore the emergence of coherence in higher dimensions, where no classical simulations are available, as well as for negative temperatures. For intermediate quench velocities, we observe a power-law behavior of the coherence length, reminiscent of the Kibble–Zurek mechanism. However, we find nonuniversal exponents that cannot be captured by this mechanism or any other known model.