  A class of tensor models were recently outlined as potentially calculable
examples of holography: their perturbative large-$N$ behavior is similar to the
Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the
sense that there is no quenched disorder averaging). These facts make them
intriguing tentative models for quantum black holes. In this note, we
explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and
study its spectral and late-time properties. We find parallels to (a single
sample of) SYK where some of these features were recently attributed to random
matrix behavior and quantum chaos. In particular, after a running time average,
the spectral form factor exhibits striking qualitative similarities to SYK. But
we also observe that even though the spectrum has a unique ground state, it has
a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If
one ignores the delta function due to the degeneracies however, there is level
repulsion in the unfolded spacing distribution hinting chaos. Furthermore, the
spectrum has gaps and is not (linearly) rigid. The system also has a spectral
mirror symmetry which we trace back to the presence of a unitary operator with
which the Hamiltonian anticommutes. We use it to argue that to the extent that
the model exhibits random matrix behavior, it is controlled not by the Dyson
ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer
classification.
