  Globally optimizing a nonconvex quadratic over the intersection of $m$ balls
in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$.
Moreover, when $m=1$, the standard semidefinite relaxation is exact. When
$m=2$, it has been shown recently that an exact relaxation can be constructed
using a disjunctive semidefinite formulation based essentially on two copies of
the $m=1$ case. However, there is no known explicit, tractable, exact convex
representation for $m \ge 3$. In this paper, we construct a new, polynomially
sized semidefinite relaxation for all $m$, which does not employ a disjunctive
approach. We show that our relaxation is exact for $m=2$. Then, for $m \ge 3$,
we demonstrate empirically that it is fast and strong compared to existing
relaxations. The key idea of the relaxation is a simple lifting of the original
problem into dimension $n+1$. Extending this construction: (i) we show that
nonconvex quadratic programming over $\|x\| \le \min \{ 1, g + h^T x \}$ has an
exact semidefinite representation; and (ii) we construct a new relaxation for
quadratic programming over the intersection of two ellipsoids, which globally
solves all instances of a benchmark collection from the literature.
