  We propose an exactly solvable model for $j_{\text{eff}}=\frac32$ local
moments on the honeycomb lattice. Our construction is guided by a symmetry
analysis and by the requirement of an exact solution in terms of a Majorana
fermion representation for multipole operators. The main interaction in the
model can be interpreted as a bond-dependent quadrupole-quadrupole interaction.
When time reversal symmetry is explicitly broken, we obtain a gapped spin
liquid with a single chiral Majorana edge mode. We also investigate another
solvable model in which the time-reversal-invariant spin liquid is coupled to
conduction electrons in a superconductor. In the presence of a Kondo-like
coupling that involves the octupole moment of the localized spins, the
itinerant electrons hybridize with the emergent Majorana fermions in the spin
liquid. This leads to spontaneous time reversal symmetry breaking and generates
odd-frequency pairing. Our results suggest that $j_{\text{eff}}=\frac32$
systems with strong quadrupole-quadrupole interactions may provide a route
towards non-Abelian quantum spin liquids and unconventional superconductivity.
