  We prove a vector-valued inequality for the Gaussian noise stability (i.e. we
prove a vector-valued Borell inequality) for Euclidean functions taking values
in the two-dimensional sphere, for all correlation parameters at most $1/10$ in
absolute value. This inequality was conjectured (for all correlation parameters
at most $1$ in absolute value) by Hwang, Neeman, Parekh, Thompson and Wright.
Such an inequality is needed to prove sharp computational hardness of the
product state Quantum MAX-CUT problem, assuming the Unique Games Conjecture. In
fact, assuming the Unique Games Conjecture, we show that the product state of
Quantum MAX-CUT is NP-hard to approximate within a multiplicative factor of
$.9859$. In contrast, a polynomial time algorithm is known with approximation
factor $.956\ldots$.
