  In this paper we study the relative Chow and $K$-stability of toric manifolds
in the toric sense. First, we give a criterion for relative $K$-stability and
instability of toric Fano manifolds in the toric sense. The reduction of
relative Chow stability on toric manifolds will be investigated using the
Hibert-Mumford criterion in two ways. One is to consider the maximal torus
action and its weight polytope. We obtain a reduction by the strategy of Ono
[Ono13], which fits into the relative GIT stability detected by Sz\'ekelyhidi.
The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to
toric degenerations following Donaldson and Ross-Thomas [D02, RT07]. As
applications of our main theorem, we partially determine the relative
$K$-stability of toric Fano threefolds and present counter-examples which are
relatively $K$-stable in the toric sense but which are asymptotically
relatively Chow unstable. In the end, we explain the erroneous parts of the
published version of this article (corresponding to Sections 1-5), which
provides some inconclusive results for relative $K$-stability in Table 6.
