The continuous random energy model (CREM) is a Gaussian process indexed by a binary tree of depth T, introduced by Derrida and Spohn (1988) and Bovier and Kurkova (2004) as a toy model of a spin glass. In this talk, I will present recent results on hardness thresholds for algorithms that search for low-energy states. I will first discuss the existence of an algorithmic hardness threshold x_*: finding a state of energy lower than -x T is possible in polynomial time if x < x_*, and takes exponential time if x > x_*, with high probability. I will then focus on the transition from polynomial to exponential complexity near the algorithmic hardness threshold, by studying the performance of a certain beam-search algorithm of beam width N depending on T — we believe this algorithm to be natural and asymptotically optimal. The algorithm turns out to be essentially equivalent to the time-inhomogeneous version of the so-called N-particle branching Brownian motion (N-BBM), which has seen a lot of interest in the last two decades. Studying the performance of the algorithm then amounts to investigating the maximal displacement at time T of the time-inhomogeneous N-BBM. In doing so, we are able to quantify precisely the nature of the transition from polynomial to exponential complexity, proving that the transition happens when the log-complexity is of the order of the cube root of T. This result appears to be the first of its kind and we believe this phenomenon to be universal in a certain sense.