We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_N$-spaces needed to represent (up to a constant $C>1$) the $M_N$-version of the $n$-dimensional operator Hilbert space $O{H}_n$ as a direct sum of copies of $M_N$. We show that, when $C$ is close to 1, this multiplicity grows as $exp\beta n{N}^2$ for...