  Bohnenblust--Hille inequalities for Boolean cubes have been proven with
dimension-free constants that grow subexponentially in the degree
\cite{defant2019fourier}. Such inequalities have found great applications in
learning low-degree Boolean functions \cite{eskenazis2022learning}. Motivated
by learning quantum observables, a qubit analogue of Bohnenblust--Hille
inequality for Boolean cubes was recently conjectured in \cite{RWZ22}. The
conjecture was resolved in \cite{CHP}. In this paper, we give a new proof of
these Bohnenblust--Hille inequalities for qubit system with constants that are
dimension-free and of exponential growth in the degree. As a consequence, we
obtain a junta theorem for low-degree polynomials. Using similar ideas, we also
study learning problems of low degree quantum observables and Bohr's radius
phenomenon on quantum Boolean cubes.
