Consider a smooth quasiprojective variety <i>X</i> equipped with a C*-action, and a regular function <i>f</i> : <i>X</i> → C which is C*-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of <i>f</i> on proper components of the critical locus of <i>f</i>, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work of Kontsevich–Soibelman, Nagao and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.