for ${{\mathcal P}}$ exists if and only if the trees in ${{\mathcal P}}$ can simultaneously be embedded into a temporal network [@humphries13 Theorem 1]. Moreover, a cherry-picking sequence for ${{\mathcal P}}$ can be exploited further to compute the minimum number of reticulation events that is needed over all temporal networks. Importantly, not every collection ${{\mathcal P}}$ is guaranteed to have a solution, i.e. there may be no cherry-picking sequence for ${{\mathcal P}}$ and, hence no temporal network that embeds all trees in ${{\mathcal P}}$. It was left as an open problem by Humphries et al. [@humphries13] to analyze the computational complexity of deciding whether or not ${{\mathcal P}}$ has a cherry-picking sequence for when $|{{\mathcal P}}|=2$.

In this paper, we make progress towards this question and show that it is NP-complete to decide if ${{\mathcal P}}$ has a cherry-picking sequence for when $|{{\mathcal P}}|\geq 8$. Translated into the language of phylogenetic networks, this result directly implies that it is computationally hard to decide if a collection of at least eight rooted binary phylogenetic trees can simultaneously be embedded into a temporal network. To establish our result, we use a reduction from a variant of the [Intermezzo]{} problem [@guttmann06]. On a 