I have read that the membership problem for CSL is PSPACE-complete but I couldn't find the proof anywhere. So I tried it myself.
Let's mark the membership problem for CSL as MEM. First I have to proof that $ MEM \in PSPACE $. This should be easy, just take Turing Machine that generate words from L in lexicographic order and check if any is the same as the input word w. We can stop with the Turing machine when we reach length w+1.
Seccond, make a reduction from some PSPACE language. Quantified Boolean formula problem (QBF) seems to be suitable for this reduction. I have seen how to make a reduction from MEM to QBF but here I need the opposite. If I had a word w, I could make a formula based on the configuration I must go through to get w and all those configurations would mean true for the QBF. The representation could be just going from the binary code to some formulas.
But when going from the opposite direction, I don't know how to make CSL work from any given formula.