  Tree tensor networks such as the tensor train format are a common tool for
high dimensional problems. The associated multivariate rank and accordant
tuples of singular values are based on different matricizations of the same
tensor. While the behavior of such is as essential as in the matrix case, here
the question about the $\textit{feasibility}$ of specific constellations
arises: which prescribed tuples can be realized as singular values of a tensor
and what is this feasible set? We first show the equivalence of the
$\textit{tensor feasibility problem (TFP)}$ to the $\textit{quantum marginal
problem (QMP)}$. In higher dimensions, in case of the tensor train (TT-)format,
the conditions for feasibility can be decoupled. By present results for three
dimensions for the QMP, it then follows that the tuples of squared, feasible
TT-singular values form polyhedral cones. We further establish a connection to
eigenvalue relations of sums of Hermitian matrices, which in turn are described
by sets of interlinked, so called $\textit{honeycombs}$, as they have been
introduced by Knutson and Tao. Besides a large class of universal, necessary
inequalities as well as the vertex description for a special, simpler instance,
we present a linear programming algorithm to check feasibility and a simple,
heuristic algorithm to construct representations of tensors with prescribed,
feasible TT-singular values in parallel.
