  In part I of this work we studied the spaces of real algebraic cycles on a
complex projective space P(V), where V carries a real structure, and completely
determined their homotopy type. We also extended some functors in K-theory to
algebraic cycles, establishing a direct relationship to characteristic classes
for the classical groups, specially Stiefel-Whitney classes. In this sequel, we
establish corresponding results in the case where V has a quaternionic
structure. The determination of the homotopy type of quaternionic algebraic
cycles is more involved than in the real case, but has a similarly simple
description. The stabilized space of quaternionic algebraic cycles admits a
nontrivial infinite loop space structure yielding, in particular, a delooping
of the total Pontrjagin class map. This stabilized space is directly related to
an extended notion of quaternionic spaces and bundles (KH-theory), in analogy
with Atiyah's real spaces and KR-theory, and the characteristic classes that we
introduce for these objects are nontrivial. The paper ends with various
examples and applications.
