  We study the kernelization of exploration problems on temporal graphs. A
temporal graph consists of a finite sequence of snapshot graphs
$\mathcal{G}=(G_1, G_2, \dots, G_L)$ that share a common vertex set but might
have different edge sets.
  The non-strict temporal exploration problem (NS-TEXP for short) introduced by
Erlebach and Spooner, asks if a single agent can visit all vertices of a given
temporal graph where the edges traversed by the agent are present in non-strict
monotonous time steps, i.e., the agent can move along the edges of a snapshot
graph with infinite speed. The exploration must at the latest be completed in
the last snapshot graph. The optimization variant of this problem is the
$k$-arb NS-TEXP problem, where the agent's task is to visit at least $k$
vertices of the temporal graph. We show that under standard computational
complexity assumptions, neither of the problems NS-TEXP nor $k$-arb NS-TEXP
allow for polynomial kernels in the standard parameters: number of vertices
$n$, lifetime $L$, number of vertices to visit $k$, and maximal number of
connected components per time step $\gamma$; as well as in the combined
parameters $L+k$, $L + \gamma$, and $k+\gamma$. On the way to establishing
these lower bounds, we answer a couple of questions left open by Erlebach and
Spooner.
  We also initiate the study of structural kernelization by identifying a new
parameter of a temporal graph
  $p(\mathcal{G}) = \sum_{i=1}^{L} (|E(G_i)|) - |V(G)| +1$. Informally, this
parameter measures how dynamic the temporal graph is. Our main algorithmic
result is the construction of a polynomial (in $p(\mathcal{G})$) kernel for the
more general Weighted $k$-arb NS-TEXP problem, where weights are assigned to
the vertices and the task is to find a temporal walk of weight at least $k$.
