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While many-body quantum physics has traditionally focused on the properties of cold matter in thermal equilibrium, emerging noisy intermediate-scale quantum (NISQ) platforms allow access to far-from-equilibrium dynamics with local space and time control over interactions. In this new era of “interactive quantum dynamics”, a key challenge is to identify universal features of non-equilibrium quantum dynamics, transport, and many-body entanglement. In this talk, I will discuss new types of non-equilibrium quantum phases of matter and phase transitions, with an emphasis on emergent classical statistical mechanics descriptions of quantum entanglement dynamics. In particular, I will focus on the recently discovered “learnability phase transitions” that occur in monitored quantum systems and separate phases characterized by the amount of quantum information that can be extracted from measurements.