  Powerful skew arithmetic circuits are introduced. These are skew arithmetic
circuits with variables, where input gates can be labelled with powers $x^n$
for binary encoded numbers $n$. It is shown that polynomial identity testing
for powerful skew arithmetic circuits belongs to $\mathsf{coRNC}^2$, which
generalizes a corresponding result for (standard) skew circuits. Two
applications of this result are presented: (i) Equivalence of
higher-dimensional straight-line programs can be tested in $\mathsf{coRNC}^2$;
this result is even new in the one-dimensional case, where the straight-line
programs produce strings. (ii) The compressed word problem (or circuit
evaluation problem) for certain wreath products of finitely generated abelian
groups belongs to $\mathsf{coRNC}^2$.
