  We introduce novel polyhedral approximation hierarchies for the cone of
nonnegative forms on the unit sphere in $\mathbb{R}^n$ and for its (dual) cone
of moments. We prove computable quantitative bounds on the speed of convergence
of such hierarchies. We also introduce a novel optimization-free algorithm for
building converging sequences of lower bounds for polynomial minimization
problems on spheres. Finally some computational results are discussed,
showcasing our implementation of these hierarchies in the programming language
Julia.
