2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144285We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.A minor change: clarified the recipe for computing the quantum homology of a symplectic toric Fano manifoldSymplectic GeometryAlgebraic Geometry53D45; 53D40; 14N35text