  A graph whose edges only appear at certain points in time is called a
temporal graph (among other names). Such a graph is temporally connected if
each ordered pair of vertices is connected by a path which traverses edges in
chronological order (i.e., a temporal path). In this paper, we consider a
simple model of random temporal graph, obtained from an Erd\H{o}s-R\'enyi
random graph $G~G_{n,p}$ by considering a random permutation $\pi$ of the edges
and interpreting the ranks in $\pi$ as presence times. Temporal reachability in
this model exhibits a surprisingly regular sequence of thresholds. In
particular, we show that at $p=\log n/n$ any fixed pair of vertices can a.a.s.
reach each other; at $2\log n/n$ at least one vertex (and in fact, any fixed
vertex) can a.a.s. reach all others; and at $3\log n/n$ all the vertices can
a.a.s. reach each other, i.e., the graph is temporally connected. Furthermore,
the graph admits a temporal spanner of size $2n+o(n)$ as soon as it becomes
temporally connected, which is nearly optimal as $2n-4$ is a lower bound. This
result is significant because temporal graphs do not admit spanners of size
$O(n)$ in general (Kempe et al, STOC 2000). In fact, they do not even admit
spanners of size $o(n^2)$ (Axiotis et al, ICALP 2016). Thus, our result implies
that the obstructions found in these works, and more generally, all
non-negligible obstructions, must be statistically insignificant: nearly
optimal spanners always exist in random temporal graphs. All the above
thresholds are sharp. Carrying the study of temporal spanners further, we show
that pivotal spanners -- i.e., spanners of size $2n-2$ made of two spanning
trees glued at a single vertex (one descending in time, the other ascending
subsequently) -- exist a.a.s. at $4\log n/n$, this threshold being also sharp.
Finally, we show that optimal spanners (of size $2n-4$) also exist a.a.s. at $p
= 4\log n/n$.
