  Witsenhausen's problem asks for the maximum fraction $\alpha_n$ of the
$n$-dimensional unit sphere that can be covered by a measurable set containing
no pairs of orthogonal points. The best upper bounds for $\alpha_n$ are given
by extensions of the Lov\'asz theta number. In this paper, optimization
hierarchies based on the Lov\'asz theta number, like the Lasserre hierarchy,
are extended to Witsenhausen's problem and similar problems. These hierarchies
are shown to converge and are used to compute the best upper bounds for
$\alpha_n$ in low dimensions.
