### Abstract:

The (multivariate) truncated moment problem is an important question in analysis with applications to mathematical physics, probability theory, etc. In the 1990’s Curto & Fialkow began their expedition of solving the truncated moment problem. It has since been studied with a variety of motivations, due to its wide ranging applications, such as to multivariate integral computations (in physics, statistics, etc.), option pricing (in finance) and optimization (in mathematics). The tracial moment problem is a non-commutative analogue of the classical moment problem. A sequence of real numbers indexed by words in non-commuting variables, invariant under cyclic permutations is called a tracial sequence. In this work we study conditions for when a tracial sequence is given by the tracial moments of some matrices, focusing on the bivariate problem in low degrees. We present sufficient conditions for Curto and Yoo’s construction of an explicit representing measure on the classical quartic binary moment problem, to hold in the tracial analogue. To each tracial sequence one can associate a multivariate Hankel matrix. If the sequence is given by moments, this matrix is positive semi-definite, but not vice-versa. In the bivariate quartic case this matrix is (7 7). It is known that positive definiteness of this matrix implies the existence of a representing measure. Here we also present a comprehensive analysis on the column relations in the Hankel matrix for lower rank cases in an attempt to find or disprove the existence of a representing measure.