In this talk I will present exact results on a novel kind of emergent random matrix universality that quantum many-body systems at infinite temperature can exhibit. Specifically, I will consider an ensemble of pure states supported on a small subsystem, generated from projective measurements of the remainder of the system in a local basis. I will show how in certain quantum chaotic dynamics, the ensemble can be provably shown to approach a universal form: it becomes uniformly distributed in Hilbert space. This goes beyond the standard paradigm of quantum thermalization, which dictates that the subsystem at late times relaxes to an ensemble of states that reproduces the expectation values of local observables in a thermal state. Instead, our results imply more generally that the distribution of quantum states themselves is indistinguishable from random states, i.e. the ensemble forms a "quantum state design" in the parlance of quantum information theory. Our work establishes bridges between quantum many-body physics, quantum information and random matrix theory, by showing that pseudo-random states can arise from isolated quantum dynamics, opening up new ways to design applications for quantum state tomography and bench-marking.