  Given a curve in quantum spin state space, we inquire what is the relation
between its geometry and the geometric phase accumulated along it. Motivated by
Mukunda and Simon's result that geodesics (in the standard Fubini-Study metric)
do not accumulate geometric phase, we find a general expression for the
derivatives (of various orders) of the geometric phase in terms of the
covariant derivatives of the curve. As an application of our results, we put
forward the brachistophase problem: given a quantum state, find the
(appropriately normalized) hamiltonian that maximizes the accumulated geometric
phase after time $\tau$ - we find an analytical solution for all spin values,
valid for small $\tau$. For example, the optimal evolution of a spin coherent
state consists of a single Majorana star separating from the rest and tracing
out a circle on the Majorana sphere.
