Research in string theory has uncovered deep connections between strongly coupled quantum systems with chaotic dynamics and the quantum physics of black hole spacetimes. The realization of these ideas for cosmological spacetimes such as de Sitter space is much less well understood. After highlighting some of the challenges for finding a quantum mechanical description of an accelerating universe, we describe some recent progress towards this goal in the form of a close link between the Sachdev-Ye-Kitaev model and quantum de Sitter spacetime in low dimensions.