The K-groups of quantum-group complex projective spaces are easy to determine, and the K-groups of multipullback quantum complex projective spaces were computed recently. The goal of this talk is to unravel the noncommutative-vector-bundle generator of the $K_0$ of quantum projective planes in both of the above cases. We achieve this goal by using pullback presentations of involved algebras, and then combining the Milnor connecting homomorphism in K-theory with Hopf-algebraic prolongation techniques applied to compact quantum principal bundles. On the way, we encounter a general principle of mapping Leavitt path algebras into pullbacks of simpler algebras. Based on joint works with Francesco D'Andrea, Carla Farsi, Tomasz Maszczyk, Mariusz Tobolski and Bartosz Zielinski.