  Abelian and non-Abelian geometric phases, known as quantum holonomies, have
attracted considerable attention in the past. Here, we show that it is possible
to associate nonequivalent holonomies to discrete sequences of subspaces in a
Hilbert space. We consider two such holonomies that arise naturally in
interferometer settings. For sequences approximating smooth paths in the base
(Grassmann) manifold, these holonomies both approach the standard holonomy. In
the one-dimensional case the two types of holonomies are Abelian and coincide
with Pancharatnam's geometric phase factor. The theory is illustrated with a
model example of projective measurements involving angular momentum coherent
states.
