  We provide a $O(\log^6 \log n)$-round randomized algorithm for distance-2
coloring in CONGEST with $\Delta^2+1$ colors. For
$\Delta\gg\operatorname{poly}\log n$, this improves exponentially on the
$O(\log\Delta+\operatorname{poly}\log\log n)$ algorithm of [Halld\'orsson,
Kuhn, Maus, Nolin, DISC'20].
  Our study is motivated by the ubiquity and hardness of local reductions in
CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos,
JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume
communication on the conflict graph, which can be simulated in LOCAL with only
constant overhead, while this may be prohibitively expensive in CONGEST. We
hope our techniques help tackle in CONGEST other coloring problems defined by
local relations.
