  Majorana stars, the antipodal directions associated with the coherent states
that are orthogonal to a spin state, provide a visualization and a geometric
understanding of the structures of general quantum states. For example, the
Berry phase of a spin-1/2 is given by half the solid angle enclosed by the
close path of its Majorana star. It is conceivable that the Berry phase of
higher spins may also be related to the geometry of the Majorana constellation.
We find that for a spin-1 state, besides the expected contributions from the
solid angles enclosed by the close paths of the two Majorana stars, the Berry
phase includes a term related to the twist of the relative position vector
around the barycenter vector of the two Majorana stars, i.e., the self-rotation
of the constellation. Interestingly, if the spin-1 state is taken as a
symmetrized two-qubit state, the extra contribution to the Berry phase is given
by the self-rotation of the Majorana constellation weighted by the quantum
entanglement of the two qubits. This discovery alludes to the relevance of the
Majorana stellar geometry in representing the deep structures of quantum states
and of quantum entanglement.
