In this talk we present an explicit KK-equivalence between the noncommutative $C^*$-algebra of continuous functions on the quantum complex projective space $C(\mathbb{C}P_q^n)$ and the commutative algebra $\mathbb{C}^{n+1}$. The construction relies on showing that the short exact sequence of $C^*$-algebras $\mathcal{K}\to C(\mathbb{C}P_q^n)\to C(\mathbb{C}P_q^{n-1})$ is split exact for all $n$. In the construction of a splitting it is crucial that $C(\mathbb{C}P_q^n)$ can be described as a graph $C^*$-algebra due to Hong and Szymański.  
This talk is based on joint work with Francesca Arici.