  In the high-temperature phase of QCD, the heavy quark momentum diffusion
constant determines, via a fluctuation-dissipation relation, how fast a heavy
quark kinetically equilibrates. This transport coefficient can be extracted
from thermal correlators via a Kubo formula. We present a lattice calculation
of the relevant Euclidean correlators in the gluon plasma, based on a recent
formulation of the problem in heavy-quark effective field theory (HQET). We
find a $\approx20%$ enhancement of the Euclidean correlator at maximal time
separation as the temperature is lowered from $6T_c$ to $2T_c$, pointing to
stronger interactions at lower temperatures. At the same time, the correlator
becomes flatter from $6T_c$ down to $2T_c$, indicating a relative shift of the
spectral weight to lower frequencies. A recent next-to-leading order
perturbative calculation of the correlator agrees with the time dependence of
the lattice data at the few-percent level. We estimate how much additional
contribution from the $\omega\lesssim T$ region of the perturbative spectral
function would be required to bring it in agreement with the lattice data at
$3.1T_c$.
