  We analyze the phase diagram of a quantum mean spherical model in terms of
the temperature $T$, a quantum parameter $g$, and the ratio $p=-J_{2}/J_{1}$,
where $J_{1}>0$ refers to ferromagnetic interactions between first-neighbor
sites along the $d$ directions of a hypercubic lattice, and $J_{2}<0$ is
associated with competing antiferromagnetic interactions between second
neighbors along $m\leq d$ directions. We regain a number of known results for
the classical version of this model, including the topology of the critical
line in the $g=0$ space, with a Lifshitz point at $p=1/4$, for $d>2$, and
closed-form expressions for the decay of the pair correlations in one
dimension. In the T=0 phase diagram, there is a critical border,
$g_{c}=g_{c}(p) $ for $d\geq2$, with a singularity at the Lifshitz point if
$d<(m+4)/2$. We also establish upper and lower critical dimensions, and analyze
the quantum critical behavior in the neighborhood of $p=1/4$.
