  We describe a nonperturbative method to compute the partition function and
correlation functions for scalar QFTs set on the $d$-dimensional sphere $S^d$.
The method relies on a Hamiltonian picture, where the theory is quantized on
$S^{d-1}$ and states evolve in time by means of a time-dependent Hamiltonian.
Crucially, the Hilbert space on $S^{d-1}$ is truncated to a finite set of
states below a cutoff. Throughout this work we focus on the $\phi^2$ and
$i\phi^3$ flows in three dimensions. In the first part of this paper we analyze
the cutoff-dependence of various observables, computing both divergent and
RG-improvement counterterms to be added to the action. Next we present
nonperturbative results for the massive scalar on $S^3$, finding good agreement
in the strong-coupling regime between numerical data and the $F$-coefficient of
the free scalar CFT. We also check that the renormalized $i \phi^3$ theory on
$S^3$ is nonperturbatively UV-finite. The scheme in question breaks the
$\mathrm{SO}(d+1)$ spacetime symmetry group of $S^d$ down to $\mathrm{SO}(d)$,
and in an example we study how the full symmetry is restored in the continuum
limit. The relation between our method and earlier work by Al. B. Zamolodchikov
involving a specific RG flow on $S^2$ is explained as well.
