  A set of pure quantum states is said to be antidistinguishable if upon
sampling one at random, there exists a measurement to perfectly determine some
state that was not sampled. We show that antidistinguishability of a set of $n$
pure states is equivalent to a property of its Gram matrix called
$(n-1)$-incoherence, thus establishing a connection with quantum resource
theories that lets us apply a wide variety of new tools to
antidistinguishability. As a particular application of our result, we present
an explicit formula (not involving any semidefinite programming) that
determines whether or not a set with a circulant Gram matrix is
antidistinguishable. We also show that if all inner products are smaller than
$\sqrt{(n-2)/(2n-2)}$ then the set must be antidistinguishable, and we show
that this bound is tight when $n \leq 4$. We also give a simpler proof that if
all the inner products are strictly larger than $(n-2)/(n-1)$, then the set
cannot be antidistinguishable, and we show that this bound is tight for all
$n$.
