Abstract: In the past 27 years there has been a lot of progress in understanding quantum gravity in anti-de Sitter spacetime. However, we live in an expanding Universe which can be approximated by a de Sitter space, which has a lot of unexpected properties. In the first part of my talk I will address some basic properties of de Sitter spacetime from the perspective of quantum dynamics. I will argue that de Sitter equilibrates in a superdiffusive manner, unlike most physical systems which equilibrate diffusively. Also I will propose that de Sitter has a minimum scrambling time of log(1/G_N ), even for perturbations introduced on the stretched horizon, indicating that de Sitter might be a fast scrambler of free bulk fields, rather than a hyperfast scrambler. In the second part of my talk I will discuss a simple quantum mechanical model, the so-called Brownian double-scaled Sachdev--Ye--Kitaev model, which shares a surprising amount of properties with de Sitter.