  We report the findings of fractional topological states in one-dimensional
periodically modulated quantum spin chains with up to third neighbor
interactions. By exact numerical studies, we demonstrate the existence of
topologically nontrivial degenerate ground states at some specific
magnetizations, which can be characterized by the nonzero-integer total Chern
numbers of the degenerate ground states and the emergence of nontrivial edge
states under open boundary conditions. We find that the low-energy excitations
obey bosonic $\nu=1/2$ fractional statistics for the spin-$1/2$ system and
$\nu=1$ non-Abelian statistics for the spin-$1$ system, respectively. The
discovered fractional quantum states provide another route to the theoretical
exploration of fractional quantum states in correlated spin systems.
