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 regimes, particularly out of equilibrium, 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. 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 spacetime duality: a mapping between unitary circuits and monitored ones based on exchanging the roles of space and time, that can be realized on NISQ processors via a generalized "quantum teleportation" protocol.