  For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for
$\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether
$\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the
word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy
is decidable. More precisely, for every fixed variety in the Trotter-Weil
Hierarchy, our approach yields an algorithm in nondeterministic logarithmic
space (NL). In addition, we provide deterministic polynomial time algorithms
which are more efficient than straightforward translations of the
NL-algorithms. As an application of our results, we show that separability by
the so-called corners of the Trotter-Weil Hierarchy is witnessed by
$\omega$-terms (this property is also known as $\omega$-reducibility). In
particular, the separation problem for the corners of the Trotter-Weil
Hierarchy is decidable.
