  We study the graph isomorphism game that arises in quantum information theory
from the perspective of bigalois extensions of compact quantum groups. We show
that every algebraic quantum isomorphism between a pair of (quantum) graphs $X$
and $Y$ arises as a quotient of a certain measured bigalois extension for the
quantum automorphism groups $G_X$ and $G_Y$ of the graphs $X$ and $Y$. In
particular, this implies that the quantum groups $G_X$ and $G_Y$ are monoidally
equivalent. We also establish a converse to this result, which says that every
compact quantum group $G$ monoidally equivalent to $G_X$ is of the form $G_Y$
for a suitably chosen quantum graph $Y$ that is quantum isomorphic to $X$. As
an application of these results, we deduce that the $\ast$-algebraic,
C$^\ast$-algebraic, and quantum commuting (qc) notions of a quantum isomorphism
between classical graphs $X$ and $Y$ all coincide. Using the notion of
equivalence for non-local games, we deduce the same result for other
synchronous non-local games, including the synBCS game and certain related
graph homomorphism games.
