  A temporal digraph ${\cal G}$ is a triple $(G, \gamma, \lambda)$ where $G$ is
a digraph, $\gamma$ is a function on $V(G)$ that tells us the timestamps when a
vertex is active, and $\lambda$ is a function on $E(G)$ that tells for each $uv
\in E(G)$ when $u$ and $v$ are linked. Given a static digraph $G$, and a subset
$R\subseteq V(G)$, a spanning branching with root $R$ is a subdigraph of $G$
that has exactly one path from $R$ to each $v\in V(G)$. In this paper, we
consider the temporal version of Edmonds' classical result about the problem of
finding $k$ edge-disjoint spanning branchings respectively rooted at given
$R_1,\cdots,R_k$. We introduce and investigate different definitions of
spanning branchings, and of edge-disjointness in the context of temporal
graphs. A branching ${\cal B}$ is vertex-spanning if the root is able to reach
each vertex $v$ of $G$ at some time where $v$ is active, while it is
temporal-spanning if $v$ can be reached from the root at every time where $v$
is active. On the other hand, two branchings ${\cal B}_1$ and ${\cal B}_2$ are
edge-disjoint if they do not use the same edge of $G$, and are
temporal-edge-disjoint if they can use the same edge of $G$ but at different
times. This lead us to four definitions of disjoint spanning branchings and we
prove that, unlike the static case, only one of these can be computed in
polynomial time, namely the temporal-edge-disjoint temporal-spanning branchings
problem, while the other versions are $\mathsf{NP}$-complete, even under very
strict assumptions.
