  The properties of the geometric phases between three quantum states are
investigated in a high-dimensional Hilbert space using the Majorana
representation of symmetric quantum states. We found that the geometric phases
between the three quantum states in an N-state quantum system can be
represented by N-1 spherical triangles on the Bloch sphere. The parameter
dependence of the geometric phase was analyzed based on this picture. We found
that the geometric phase exhibits rich nonlinear behavior in a high-dimensional
Hilbert space.
