  We study both classical and quantum algorithms to solve a hard optimization
problem, namely 3-XORSAT on 3-regular random graphs. By introducing a new
quasi-greedy algorithm that is not allowed to jump over large energy barriers,
we show that the problem hardness is mainly due to entropic barriers. We study,
both analytically and numerically, several optimization algorithms, finding
that entropic barriers affect in a similar way classical local algorithms and
quantum annealing. For the adiabatic algorithm, the difficulty we identify is
distinct from that of tunnelling under large barriers, but does, nonetheless,
give rise to exponential running (annealing) times.
