  Given an undirected graph $G=(V,E)$ the NP-hard Strong Triadic Closure (STC)
problem asks for a labeling of the edges as \emph{weak} and \emph{strong} such
that at most $k$ edges are weak and for each induced $P_3$ in $G$ at least one
edge is weak. In this work, we study the following generalizations of STC with
$c$ different strong edge colors. In Multi-STC an induced $P_3$ may receive two
strong labels as long as they are different. In Edge-List Multi-STC and
Vertex-List Multi-STC we may additionally restrict the set of permitted colors
for each edge of $G$. We show that, under the Exponential Time Hypothesis
(ETH), Edge-List Multi-STC and Vertex-List Multi-STC cannot be solved in time
$2^{o(|V|^2)}$. We then proceed with a parameterized complexity analysis in
which we extend previous fixed-parameter tractability results and
kernelizations for STC [Golovach et al., Algorithmica '20, Gr\"uttemeier and
Komusiewicz, Algorithmica '20] to the three variants with multiple edge colors
or outline the limits of such an extension.
