Theorem 6 here https://arxiv.org/pdf/quant-ph/0005018.pdf 
is one relationship between QC and the von Neumann entropy (S).https://arxiv.org/abs/0712.4377 is another good reference.
My very rough intuition for why the link between S and QC is not as strong as that between Shannon entropy (H) and Kolmogorov Complexity (K) is: If bit-strings, $x$, are drawn according to some distribution $p$ then $H(p)$ is the expected value of $K(x)$. So, both $K$ and $H$ measure, in some sense, the complexity. Now let $\mu$ and $\rho$ be a density matrices with the same eigenvalues but with different eigenvectors. Suppose the eigenvectors of $\mu$ are more complex (in the sense of $QC$) than those of $\rho$. I.e., $QC(\rho)<QC(\mu)$. Now, $S(\rho)=S(\mu)$ as S only depends on the eigenvalues, not on the eigenvectors. So, by being blind to the complexity of the eigenvectors, S becomes unrelated to the QC.