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.
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