To investigate the performance of quantum information tasks on networks whose topology changes in time, we study the spatial search algorithm by continuous time quantum walk to find a marked node on a random temporal network. We consider a network of n nodes constituted by a time-ordered sequence of Erdös-Rényi random graphs G(n,p), where p is the probability that any two given nodes are connected: after every time interval τ, a new graph G(n,p) replaces the previous one. We prove analytically

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... hat for any given p, there is always a range of values of τ for which the running time of the algorithm is optimal, i.e. O(√(n)), even when search on the individual static graphs constituting the temporal network is sub-optimal. On the other hand, there are regimes of τ where the algorithm is sub-optimal even when each of the underlying static graphs are sufficiently connected to perform optimal search on them. From this first study of quantum spatial search on a time-dependent network, it emerges that the non-trivial interplay between temporality and connectivity is key to the algorithmic performance. Moreover, our work can be extended to establish high-fidelity qubit transfer between any two nodes of the network. Overall, our findings show that one can exploit temporality to achieve optimal quantum information tasks on dynamical random networks.