  Positive topological entropy and distributional chaos are characterized for
hereditary shifts. A hereditary shift has positive topological entropy if and
only if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it is
not uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it is
not proximal (has more than one minimal set). As every spacing shift and every
beta shift is hereditary the results apply to those classes of shifts. Two open
problems on topological entropy and distributional chaos of spacing shifts from
an article of Banks et al. are solved thanks to this characterization.
Moreover, it is shown that a spacing shift $\Omega_P$ has positive topological
entropy if and only if $\mathbb{N}\setminus P$ is a set of Poincar\'{e}
recurrence. Using a result of K\v{r}\'{\i}\v{z} an example of a proximal
spacing shift with positive entropy is constructed. Connections between spacing
shifts and difference sets are revealed and the methods of this paper are used
to obtain new proofs of some results on difference sets.
