  The symmetry SU(2) and its geometric Bloch Sphere rendering are familiar for
a qubit (spin-1/2) but extension of symmetries and geometries have been
investigated far less for multiple qubits, even just a pair of them, that are
central to quantum information. In the last two decades, two different
approaches with independent starting points and motivations have come together
for this purpose. One was to develop the unitary time evolution of two or more
qubits for studying quantum correlations, exploiting the relevant Lie algebras
and especially sub-algebras of the Hamiltonians involved, and arriving at
connections to finite projective geometries and combinatorial designs.
Independently, geometers studying projective ring lines and associated finite
geometries have come to parallel conclusions. This review brings together both
the Lie algebraic and group representation perspective of quantum physics and
the geometric algebraic one, along with connections to complex quaternions.
Together, all this may be seen as further development of Felix Klein's Erlangen
Program for symmetries and geometries. In particular, the fifteen generators of
the continuous SU(4) Lie group for two-qubits can be placed in one-to-one
correspondence with finite projective geometries, combinatorial Steiner
designs, and finite quaternionic groups. The very different perspectives may
provide further insight into problems in quantum information. Extensions are
considered for multiple qubits and higher spin or higher dimensional qudits.
