  We consider the LOCAL model of distributed computing, where in a single round
of communication each node can send to each of its neighbors a message of an
arbitrary size. It is know that, classically, the round complexity of
3-coloring an $n$-node ring is $\Theta(\log^*\!n)$. In the case where
communication is quantum, only trivial bounds were known: at least some
communication must take place.
  We study distributed algorithms for coloring the ring that perform only a
single round of one-way communication. Classically, such limited communication
is already known to reduce the number of required colors from $\Theta(n)$, when
there is no communication, to $\Theta(\log n)$. In this work, we show that the
probability of any quantum single-round one-way distributed algorithm to output
a proper $3$-coloring is exponentially small in $n$.
