  Geodesics are the shortest curves between any two points on a given surface.
Geodesics in the state space of quantum systems play an important role in the
theory of geometric phases, as these are also the curves along which the
acquired geometric phase is zero. Null phase curves (NPCs) are the
generalization of the geodesics, which are defined as the curves along which
the acquired geometric phase is zero even though they need not be the shortest
curves between two points. Here we present a geometric decomposition of
geodesics and NPCs in higher-dimensional state space, which allows
understanding the intrinsic symmetries of these curves. We use Majorana star
representation to decompose a geodesic in the $n$-dimensional Hilbert space to
$n-1$ curves on the Bloch sphere and show that all the $n-1$ curves are
circular segments with specific properties that are determined by the inner
product of the end states connected by the given geodesic. We also propose a
method to construct infinitely many NPCs between any two arbitrary states for
$(n>2)$-dimensional Hilbert space using our geometric decomposition.
