Let $f:\{-1,1\}^n \to$ R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in $\{-1,1\}$.

We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most $k$ terms must either be Boolean, or output values different than $-1$ or $1$ for a fraction of at least $2/(k+2)^2$ of its domain.

It follows that given oracle access to $f$, together with the guarantee that its representation as a multilinear polynomial has at most $k$ terms, one can test Booleanity using $O(k^2)$ queries. We show an $\Omega(k)$ queries lower bound for this problem.

Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle