Abstract: Rapid advancement in neutron-star observations allows unprecedented empirical access to cold, ultra-dense QCD matter, complementing collider experiments. The combination of these observations with theoretical calculations reveals previously inaccessible features of the equation of state and the phase diagram of QCD. In this talk, I demonstrate how perturbative-QCD calculations at asymptotically high densities robustly constrain the equation of state at neutron-star densities using a new method solely based on causality and stability. I confront these calculations with neutron-star observations in a Gaussian-process-based Bayesian framework and demonstrate that the perturbative-QCD calculations offer significant and nontrivial information, going beyond that which is obtainable from current observations. The main effect of the QCD input is to soften the equation of state at high densities, supporting the hypothesis that most massive neutron stars have quark matter cores.