  Geometric phases are important in quantum physics and now central to fault
tolerant quantum computation. For spin-1/2 and SU(2), the Bloch sphere $S^2$,
together with a U(1) phase, provides a complete SU(2) description. We
generalize to $N$-level systems and SU($N$) in terms of a $2(N-1)$-dimensional
base space and reduction to a ($N$-1)-level problem, paralleling closely the
two-dimensional case. This iteratively solves the time evolution of an
$N$-level system and gives ($N$-1) geometric phases explicitly. A complete
analytical construction of a S^4 Bloch-like sphere for two qubits is given for
the Spin(5) or SO(5) subgroup of SU(4).
