  We demonstrate that ab-initio calculations in QCD at high densities offer
significant and nontrivial information about the equation of state of matter in
the cores of neutron stars, going beyond that which is obtainable from current
astrophysical observations. We do so by extrapolating the equation of state to
neutron-star densities using a Gaussian process and conditioning it
sequentially with astrophysical observations and QCD input. Using our recent
work, imposing the latter does not require an extrapolation to asymptotically
high density. We find the QCD input to be complementary to the astrophysical
observations, offering strong additional constraints at the highest densities
reached in the cores of neutron stars; with the QCD input, the equation of
state is no longer prior dominated at any density. The QCD input reduces the
pressure and speed of sound at high densities, and it predicts that binary
collisions of equal-mass neutron stars will produce a black hole with greater
than $95\%$ ($68\%$) credence for masses $M \geq 1.38 M_\odot$ ($M \geq 1.25
M_\odot$). We provide a Python implementation of the QCD likelihood function so
that it can be conveniently used within other inference setups.
