  We present a theory of the high-spin generalization of topological insulators
and their doped superconducting states. The higher-spin topological insulators
involve a pair of $J=3/2$ bands with opposite parity, and are characterized by
their band inversion. The low-energy effective theory reveals that the
topological insulators host four different phases characterized by mirror Chern
numbers, at which boundaries two different patterns of bulk Dirac points
appear. For the carrier-doped case, it is shown that the system may host unique
unconventional superconductivity because of its high-spin nature and additional
orbital degrees of freedom intrinsic to topological insulators. The
superconducting critical temperature is evaluated by using density-density
pairing interactions, and odd-parity Cooper pairs are shown to be naturally
realized in the presence of interorbital pairing interaction. It is observed
that even the simplest spin 0 odd-parity pairing state exhibits a novel class
of topological superconductivity---high winding topological superconductivity.
We also discuss the experimental signals of high winding topological
superconductivity in the case of the antiperovskite superconductor
Sr$_{3-x}$SnO.
