Rapid progress in quantum computing technologies is ushering in a new era for quantum many-body physics. Today’s noisy, intermediate-scale quantum (NISQ) devices, while still far from fault-tolerant quantum computers, are exceptional laboratory systems, with large many-body Hilbert spaces and unprecedented capabilities for control and measurement. This allows the exploration of quantum dynamics in new far-from-equilibrium regimes, and motivates new paradigms of phase structure. In this talk I will focus on two such paradigms: eigenstate-ordered phases in periodically driven systems, and entanglement phases in “monitored” systems, whose dynamics include projective measurements alongside unitary operations. As an example of the former, I will discuss the realization of a “discrete time crystal” (DTC) on Google Quantum AI’s Sycamore processor, focusing on the conceptual challenges involved in detecting the DTC’s signature eigenstate order despite intrinsic limitations of NISQ hardware. I will then present a new window into measurement-induced entanglement phases based on the idea of space-time duality: a transformation that relates unitary and monitored circuits by exchanging the roles of space and time in the dynamics, which can be implemented on digital quantum simulators through a generalized “quantum teleportation” protocol.