  The quantum three-rotor problem concerns the dynamics of three equally
massive particles moving on a circle subject to pairwise attractive cosine
potentials and can model coupled Josephson junctions. Classically, it displays
order-chaos-order behavior with increasing energy. The quantum system admits a
dimensionless coupling with semiclassical behavior at strong coupling. We study
stationary states with periodic `relative' wave functions. Perturbative and
harmonic approximations capture the spectrum at weak coupling and that of
low-lying states at strong coupling. More generally, the cumulative
distribution of energy levels obtained by numerical diagonalization is
well-described by a Weyl-like semiclassical estimate. However, the system has
an $S_3 \times \mathbb{Z}_2$ symmetry that is obscured when working with
relative angles. By exploiting a basis for invariant states, we obtain the
purified spectrum in the identity representation. To uncover universal quantum
hallmarks of chaos, we partition the spectrum into energy windows where the
classical motion is regular, mixed or chaotic and unfold each separately. At
strong coupling, we find striking signatures of transitions between regularity
and chaos: spacing distributions morph from Poisson to Wigner-Dyson while the
number variance shifts from linear to logarithmic behavior at small lengths.
Some nonuniversal features are also examined. For instance, the number variance
saturates and oscillates at large lengths for strong coupling while deviations
from Poisson spacings at asymptotically low and high energies are
well-explained by purified quantum harmonic and free-rotor spectra at strong
and weak coupling. Interestingly, the degeneracy of free-rotor levels admits an
elegant formula that we deduce using properties of Eisenstein primes.
