In this talk, we present some results on the classification of noncommutative
vector bundles over quantum odd-dimensional spheres up to isomorphism or
equivalently the classification of projections over their C*-algebras up to
equivalence. We also identify as elementary concrete projections those quantum
line bundles over the associated quantum complex projective spaces naturally
arising from the canonically associated Hopf fibrations. We cover two
well-known different versions of quantum odd-dimensional spheres, one
constructed from quantum SU(n) and the other from multipullback process. Some
joint work with D'Andrea, Hajac, Maszczyk, and Zieli\'{n}ski will be also
presented, including a quantum version for two Atiyah-Todd equalities in the
classical Adams theory about the K$_{0}$-ring of complex projective spaces.