It is well known that $\mathsf{NL} \subseteq \mathsf{NC} \subseteq \mathsf{P}$, both inclusions conjectured to be proper. On the other hand $\mathsf{NP} \supseteq \mathsf{P}$, also probably a proper inclusion. There are classes naturally interpolating between $\mathsf{NL}$ and $\mathsf{NP}$, namely $\mathsf{NSC}^k$, languages decidable by non-deterministic Turing machines in simultaneous polynomial time complexity and $O(\log^k n)$ space complexity. How big are these classes? We have $\mathsf{NSC}^k \subseteq \mathsf{polyL}$ since $NSPACE(O(\log^k n)) \subseteq DSPACE(O(\log^{2k} n))$Do we have $\mathsf{NSC}^k \subseteq \mathsf{NC}$? $\mathsf{NSC}^k \subseteq \mathsf{P}$?  $\mathsf{NSC}^k = \mathsf{NP}$? What is is known about these questions for different values of $k$ > 1?
How big is NSC^k?
cc.complexity theory;space bounded;nondeterminism