  We work on the classification of isomorphism classes of finitely generated
projective modules over the C*-algebras $C\left( \mathbb{P}^{n}\left(
\mathcal{T}\right) \right) $ and $C\left( \mathbb{S}_{H}^{2n+1}\right) $ of the
quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T} \right) $
and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles
$L_{k}$ over $\mathbb{P}^{n}\left( \mathcal{T}\right) $, studied by Hajac and
collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault
to the study of C*-algebraic structure, we analyze $C\left(
\mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, $C\left(
\mathbb{S}_{H}^{2n+1}\right) $, and $L_{k}$ in the context of groupoid
C*-algebras, and then apply Rieffel's stable rank results to show that all
finitely generated projective modules over $C\left( \mathbb{S}_{H}
^{2n+1}\right) $ of rank higher than $\left\lfloor \frac{n}{2}\right\rfloor +3$
are free modules. Furthermore, besides identifying a large portion of the
positive cone of the $K_{0}$-group of $C\left( \mathbb{P}^{n}\left(
\mathcal{T}\right) \right) $, we also explicitly identify $L_{k}$ with concrete
representative elementary projections over $C\left( \mathbb{P} ^{n}\left(
\mathcal{T}\right) \right) $.
