Lieb and Seiringer stated in their reformulation of the Bessis–Moussa–Villani conjecture that all coefficients of the polynomial p(t) = tr((A +tr B) ^m ) are non-negative whenever A and B are any two positive semidefinite matrices of the same size. We will show that for all m∈ℕ the coefficient of t ⁴ in p(t) is non-negative, using a connection to sums of Hermitian squares of non-commutative polynomials which has been established by Klep and Schweighofer. This implies by a well-known result of Hillar that the coefficients of t ^k are non-negative for 0 ≤ k ≤ 4, and by symmetry as well for m ≥ k ≥ m − 4.

We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefinite and of finite rank. A truncated tracial sequence allows for such a representation if and only if one of its extensions admits a flat extension. Finally, we apply the theory via duality to investigate trace-positive polynomials in non-commuting variables.