Rapid progress in quantum computing technologies is ushering in a new era for quantum many-body physics. Today's noisy, intermediate-scale quantum 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 one such paradigm: entanglement phases in monitored systems, whose dynamics include projective measurements alongside unitary operations. I will review the surprising behavior of these systems, focusing on the existence of an entangling phase supported by a dynamically-generated quantum error correcting code. I will then present a new window into this physics 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.