  Global phase factors of topological origin, resulting from cyclic local
$\rm{SU}$ evolution, called topological phases, were first described in [Phys.
Rev. Lett. {\bf 90}, 230403 (2003)], in the case of entangled qubit pairs. In
this paper we investigate topological phases in multi-qubit systems as the
result of cyclic local $\rm{SU(2)}$ evolution. These phases originate from the
topological structure of the local $\rm{SU(2)}$-orbits and are an attribute of
most entangled multi-qubit systems. We discuss the relation between topological
phases and SLOCC-invariant polynomials and give examples where topological
phases appear. A general method to find the values of the topological phases in
an $n$-qubit system is described and a complete list of these phases for up to
seven qubits is given.
