De Sitter space, complexity, and the double-scaled SYK model

The cosmological horizon has an associated entropy suggesting that it might encode a quantum mechanical system on its surface. This has motivated extending the principles of the anti-de Sitter (AdS) space/ conformal field theory (CFT) correspondence to static patch holography. Some of our understanding of the AdS/CFT holography has been catalyzed by exploring the geometric duals to quantum information observables in the CFT side of the correspondence. In particular, holographic complexity captures the evolution of the black hole interior, which is conjectured to be dual to a measure of the difficulty in preparing a quantum circuit on the CFT side. Motivated by these developments, we will propose complexity observables defined in geometric terms for an asymptotically dS spacetime and its putative quantum mechanical dual system. To sharply define complexity observables in the static patch of dS space, they will be anchored to a time-like Dirichlet surface where the dual theory is located, called the stretched horizon. While static patch holography needs much more development in order to reach a level of understanding of the likes of the AdS/CFT correspondence, we hope that the work reported in this thesis allows for crucial steps in this direction.