  We investigate possible quantifications of Banach-Saks sets and weak
Banach-Saks sets of higher orders and their relations to other quantities. We
prove a quantitative version of the characterization of weak $\xi$-Banach-Saks
sets using $\ell_1^{\xi+1}$-spreading models and a quantitative version of the
relation of $\xi$-Banach-Saks sets, weak $\xi$-Banach-Saks sets, norm
compactness and weak compactness. We further introduce a new measure of weak
compactness. Finally, we provide some examples showing the limitations of these
quantifications.
