Known properties of a specific class of quantum states
M. G. Parker and V. Rijmen have studied the quantum entanglement of binary and bipolar sequences in arXiv:quant-ph/0107106. I remember reading their paper long ago. They use a lot of terminology from coding theory, so now by just skimming through, I can't understand exactly what their result is, but you can have a deeper look.  

Notice also that graph states are a special case of your states. In particular, $\psi$ is a graph state if and only if the sequence $x$ is generated by a quadratic polynomial in GF(2).