Correlation among Psuedo Random Sequences generated from seeds which are correlated
Caveat: LFSRs outputs are vulnerable to known plaintext attacks using the Berlekamp Massey algorithm.

Given a maximal length sequence $s(t)$ of period $2^n-1$ generated by a primitive LFSR, all its windows of length $\ell \geq n$ are unique so partial period correlation of such lengths can uniquely determine the phase, thus the seed.

All primitive LFSR sequences of this length are obtainable by regular decimations of this sequence, i.e., $s(dt)$ where $gcd(2^n-1,d)=1.$

So this is an exploitable correlation.

Another relevant property is the shift and add property

$$s(t+k)\oplus s(t)=s(t+z(k))$$

that comes from the fact that the maximal length sequence and its cyclic shifts together with the all zero vector form a linear code. This is another source of correlation.Here $$a^{z(k)} =1+ a^k$$ defines the so called Zech logarithm $z(\cdot)$, if we use the trace representation $$s(t)=Tr(a^t)$$ with $a \in GF(2^n).$

The book by Golomb and Gong and a survey by Sarwate and Pursley in the Proceedings of the IEEE are good places to start.