Known properties of a specific class of quantum states Recently, I have been studying a quantum protocol for the "Hidden Matching" problem that makes use of states that can be expressed as

$|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^n (-1)^{x_i}|i\rangle$,

where $x_i\in \{0,1\}$ and $n$ is a power of 2. Note that instead of an $n$-level system we can consider these as states of $\log_2 n$ qubits. I suspect these states to have appeared in other contexts as they are a straightforward and useful way to map an $n$ bit string into an exponentially smaller amount of qubits. Hence I am wondering: have the properties of these types of states been already extensively studied?

I am particularly interested in how the reduced density matrix of each qubit and the entanglement of the state depend on the bit values $x_i$.

Thank you!
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).