  A temporal graph is an undirected graph $G=(V,E)$ along with a function that
assigns a time-label to each edge in $E$. A path in $G$ with non-decreasing
time-labels is called temporal path and the distance from $u$ to $v$ is the
minimum length (i.e., the number of edges) of a temporal path from $u$ to $v$.
A temporal $\alpha$-spanner of $G$ is a (temporal) subgraph $H$ that preserves
the distances between any pair of vertices in $V$, up to a multiplicative
stretch factor of $\alpha$. The size of $H$ is the number of its edges.
  In this work we study the size-stretch trade-offs of temporal spanners. We
show that temporal cliques always admit a temporal $(2k-1)-$spanner with
$\tilde{O}(kn^{1+\frac{1}{k}})$ edges, where $k>1$ is an integer parameter of
choice. Choosing $k=\lfloor\log n\rfloor$, we obtain a temporal $O(\log
n)$-spanner with $\tilde{O}(n)$ edges that has almost the same size (up to
logarithmic factors) as the temporal spanner in [Casteigts et al., JCSS 2021]
which only preserves temporal connectivity.
  We then consider general temporal graphs. Since $\Omega(n^2)$ edges might be
needed by any connectivity-preserving temporal subgraph [Axiotis et al.,
ICALP'16], we focus on approximating distances from a single source. We show
that $\tilde{O}(n/\log(1+\varepsilon))$ edges suffice to obtain a stretch of
$(1+\varepsilon)$, for any small $\varepsilon>0$. This result is essentially
tight since there are temporal graphs for which any temporal subgraph
preserving exact distances from a single-source must use $\Omega(n^2)$ edges.
We extend our analysis to prove an upper bound of $\tilde{O}(n^2/\beta)$ on the
size of any temporal $\beta$-additive spanner, which is tight up to
polylogarithmic factors.
  Finally, we investigate how the lifetime of $G$, i.e., the number of its
distinct time-labels, affects the trade-off between the size and the stretch of
a temporal spanner.
