We present a mathematical framework to study geometry and topology of quotients for multi-particle quantum systems. In particular, we are interested in geometrical and topological properties of abelian symplectic reduced spaces of pure multipartite states, as complex projective spaces, which are acted upon in a Hamiltonian fashion by maximal tori of the semisimple compact Local Unitary Lie groups. We discuss that the existing geometrical methods equip us with a powerful set of tools to compute topological invariants for these reduced spaces. More precisely, given the components in the moment (Kirwan) polytope for multi-qubits, we utilize a recursive wall-crossing formula for the Poincaré polynomials and Euler characteristics of abelian symplectic quotients and as some examples we elaborate the procedure for quantum systems with two and three qubits in their pure states and propose an algorithm for a general r-qubits.