  Chaos and complexity entail an entropic and computational obstruction to
describing a system, and thus are intrinsically difficult to characterize. In
this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE)
Hamiltonians and analytically compute out-of-time-ordered correlation functions
(OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and
circuit complexity. While our random matrix analysis gives a qualitatively
correct prediction of the late-time behavior of chaotic systems, we find
unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling
time and the apparent breakdown of spatial and temporal locality. The salient
feature of GUE Hamiltonians which gives us computational traction is the
Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics
look the same in any basis. Motivated by this property of the GUE, we introduce
$k$-invariance as a precise definition of what it means for the dynamics of a
quantum system to be described by random matrix theory. We envision that the
dynamical onset of approximate $k$-invariance will be a useful tool for
capturing the transition from early-time chaos, as seen by OTOCs, to late-time
chaos, as seen by random matrix theory.
