  The Gaussian noise stability of a function $f:\mathbb{R}^n \to \{-1, 1\}$ is
the expected value of $f(\boldsymbol{x}) \cdot f(\boldsymbol{y})$ over
$\rho$-correlated Gaussian random variables $\boldsymbol{x}$ and
$\boldsymbol{y}$. Borell's inequality states that for $-1 \leq \rho \leq 0$,
this is minimized by the halfspace $f(x) = \mathrm{sign}(x_1)$. In this work,
we generalize this result to hold for functions $f:\mathbb{R}^n \to S^{k-1}$
which output $k$-dimensional unit vectors. Our main conjecture, which we call
the $\textit{vector-valued Borell's inequality}$, asserts that the expected
value of $\langle f(\boldsymbol{x}), f(\boldsymbol{y})\rangle$ is minimized by
the function $f(x) = x_{\leq k} / \Vert x_{\leq k} \Vert$, where $x_{\leq k} =
(x_1, \ldots, x_k)$. We give several pieces of evidence in favor of this
conjecture, including a proof that it does indeed hold in the special case of
$n = k$.
  As an application of this conjecture, we show that it implies several
hardness of approximation results for a special case of the local Hamiltonian
problem related to the anti-ferromagnetic Heisenberg model known as Quantum
Max-Cut. This can be viewed as a natural quantum analogue of the classical
Max-Cut problem and has been proposed as a useful testbed for developing
algorithms. We show the following, assuming our conjecture:
  (1) The integrality gap of the basic SDP is $0.498$, matching an existing
rounding algorithm. Combined with existing results, this shows that the basic
SDP does not achieve the optimal approximation ratio.
  (2) It is Unique Games-hard (UG-hard) to compute a
$(0.956+\varepsilon)$-approximation to the value of the best product state,
matching an existing approximation algorithm.
  (3) It is UG-hard to compute a $(0.956+\varepsilon)$-approximation to the
value of the best (possibly entangled) state.
