  We study how quantum states are scrambled via braiding in systems of
non-Abelian anyons through the lens of entanglement spectrum statistics. In
particular, we focus on the degree of scrambling, defined as the randomness
produced by braiding, at the same amount of entanglement entropy. To quantify
the degree of randomness, we define a distance between the entanglement
spectrum level spacing distribution of a state evolved under random braids and
that of a Haar-random state, using the Kullback-Leibler divergence
$D_{\mathrm{KL}}$. We study $D_{\mathrm{KL}}$ numerically for random braids of
Majorana fermions (supplemented with random local four-body interactions) and
Fibonacci anyons. For comparison, we also obtain $D_{\mathrm{KL}}$ for the
Sachdev-Ye-Kitaev model of Majorana fermions with all-to-all interactions,
random unitary circuits built out of (a) Hadamard (H), $\pi/8$ (T), and CNOT
gates, and (b) random unitary circuits built out of two-qubit Haar-random
unitaries. To compare the degree of randomness that different systems produce
beyond entanglement entropy, we look at $D_{\mathrm{KL}}$ as a function of the
Page limit-normalized entanglement entropy $S/S_{\mathrm{max}}$. Our results
reveal a hierarchy of scrambling among various models --- even for the same
amount of entanglement entropy --- at intermediate times, whereas all models
exhibit the same late-time behavior. In particular, we find that braiding of
Fibonacci anyons randomizes initial product states more efficiently than the
universal H+T+CNOT set.
