  Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the
positivity of quantum cluster coefficients for all skew-symmetric quantum
cluster algebras, via a proof of a conjecture first suggested by Kontsevich on
the purity of mixed Hodge structures arising in the theory of cluster mutation
of spherical collections in 3-Calabi-Yau categories. The result implies
positivity, as well as the stronger Lefschetz property conjectured by Efimov,
and also the classical positivity conjecture of Fomin and Zelevinsky, recently
proved by Lee and Schiffler. Closely related to these results is a categorified
"no exotics" type theorem for cohomological Donaldson-Thomas invariants, which
we discuss and prove in the appendix.
