  Quantum states are represented by positive semidefinite Hermitian operators
with unit trace, known as density matrices. An important subset of quantum
states is that of separable states, the complement of which is the subset of
\textit{entangled} states. We show that the problem of deciding whether a
quantum state is entangled can be seen as a moment problem in real analysis.
Only a small number of such moments are accessible experimentally, and so in
practice the question of quantum entanglement of a many-body system can be
reduced to a truncated moment problem. By considering quantum entanglement of
$n$ identical atoms we arrive at the truncated moment problem defined for
symmetric measures over a product of $n$ copies of unit balls in
$\mathbb{R}^d$. We work with moments up to degree $2$. We derive necessary and
sufficient conditions for belonging to the moment cone, which can be expressed
via a linear matrix inequality of size at most $2d+2$, which is independent of
$n$. The linear matrix inequalities can be converted into a set of explicit
semialgebraic inequalities giving necessary and sufficient conditions for
membership in the moment cone, and show that the two conditions approach each
other in the limit of large $n$. The inequalities are derived via considering
the dual cone of nonnegative polynomials, and its sum-of-squares relaxation. We
show that the sum-of-squares relaxation of the dual cone is asymptotically
exact, and using symmetry reduction techniques (J. Pure Appl. Algebra 192, no.
1-3, 95, arXiv:1205.3102), it can be written as a small linear matrix
inequality of size at most $2d+2$, which is independent of $n$. For the cone of
symmetric nonnegative polynomials with the relevant support we also prove an
analogue of the half-degree principle for globally nonnegative symmetric
polynomials (J. Math. Anal. Appl. 284, no. 1, 174, J. Pure Appl. Algebra 216,
no. 4, 850).
