  We study the compactness in $L^{1}_{loc}$ of the semigroup mapping $(S_t)_{t
> 0}$ defining entropy weak solutions of general hyperbolic systems of
conservation laws in one space dimension. We establish a lower estimate for the
Kolmogorov $\varepsilon$-entropy of the image through the mapping $S_t$ of
bounded sets in $L^{1}\cap L^\infty$, which is of the same order
$1/\varepsilon$ as the ones established by the authors for scalar conservation
laws. We also provide an upper estimate of order $1/\varepsilon$ for the
Kolmogorov $\varepsilon$-entropy of such sets in the case of Temple systems
with genuinely nonlinear characteristic families, that extends the same type of
estimate derived by De Lellis and Golse for scalar conservation laws with
convex flux. As suggested by Lax, these quantitative compactness estimates
could provide a measure of the order of "resolution" of the numerical methods
implemented for these equations.
