Quantum mechanics exhibits a stark dichotomy between unitary time-evolution and measurement. These aspects are further contrasted by the fact that traditional many-body quantum theory is developed solely based on unitary aspects. In this talk, I will explore a fruitful synergy that emerges from the interplay between many-body quantum physics and the non-equilibrium quantum dynamics that arises from measurements. In particular, I will show how measurements can be used to circumvent fundamental constraints imposed by unitary dynamics and efficiently prepare a large class of topological phases of matter. In addition to discovering a new hierarchy of many-body quantum states unseen in the unitary setup and a surprising connection to the unsolvability of the quintic polynomial, our studies also yield practical protocols for quantum processors that led to the first unambiguous observation of non-Abelian anyons.