  We present an exactly solvable spin-3/2 model defined on a pentacoordinated
three-dimensional graphite lattice, which realizes a novel quantum spin liquid
with second-order topology. The exact solutions are described by Majorana
fermions coupled to a background $\mathbb{Z}_2$ gauge field, whose ground-state
flux configuration gives rise to an emergent off-centered spacetime inversion
symmetry. The symmetry protects topologically nontrivial band structures for
the Majorana fermions, particularly including nodal-line semimetal phases with
twofold topological charges: the second Stiefel-Whitney number and the
quantized Berry phase. The former leads to rich topological phenomena on the
system boundaries. There are two nodal-line semimetal phases hosting hinge
Fermi arcs located on different hinges, and they are separated by a critical
Dirac semimetal state with surface helical Fermi arcs. In addition, we show
that rich symmetry/topology can be explored in our model by simply varying the
lattice or interaction arrangement. As an example, we discuss how to achieve a
topological gapped phase with surface Dirac points.
