  The set of two-body reduced states of translation invariant, infinite quantum
spin chains can be approximated from inside and outside using matrix product
states and marginals of finite systems, respectively. These lead to hierarchies
of algebraic approximations that become tight only in the limit of infinitely
many auxiliary variables. We show that this is necessarily so for any algebraic
ansatz by proving that the set of reduced states is not semialgebraic. We also
provide evidence that additional elementary transcendental functions cannot
lead to a finitary description.
