On the Fourier spectrum of functions on Boolean cubes

Let $f$ be a real-valued, degree-$d$ Boolean function defined on the $n$-dimensional Boolean cube $\{\pm 1\}^{n}$, and $f(x) = \sum_{S \subset \{1,\ldots,d\}} \widehat{f}(S) \prod_{k \in S} x_k$ its Fourier-Walsh expansion. The main result states that there is an absolute constant $C >0$ such that the $\ell_{2d/(d+1)}$-sum of the Fourier coefficients of $f:\{\pm 1\}^{n} \rightarrow [-1,1]$ is bounded by $\leq C^{\sqrt{d \log d}}$. It was recently proved that a similar result holds for complex-valued polynomials on the $n$-dimensional poly torus $\mathbb{T}^n$, but that in contrast to this a replacement of the $n$-dimensional torus $\mathbb{T}^n$ by $n$-dimensional cube $[-1, 1]^n$ leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.