  We quantify the representational power of matrix product states (MPS) for
entangled qubit systems by giving polynomial expressions in a pure quantum
state's amplitudes which hold if and only if the state is a translation
invariant matrix product state or a limit of such states. For systems with few
qubits, we give these equations explicitly, considering both periodic and open
boundary conditions. Using the classical theory of trace varieties and trace
algebras, we explain the relationship between MPS and hidden Markov models and
exploit this relationship to derive useful parameterizations of MPS. We make
four conjectures on the identifiability of MPS parameters.
