TITLE: Results relying on higher derived algebraic geometry
QUESTION [10 upvotes]: Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-Artin stacks for $n>1$? We are using Toen--Vezzosi terminology.
EDIT: the $n$-Artin stack in question should not to be obtained as the quotient of the constant groupoid associated to a $(n-1)$-Artin stack. I did not specify this initially, my fault, but I think the requirement is pretty natural. Neither of the two answers I can see at the time of the edit address this point.

REPLY [7 votes]: The derived moduli stack of perfect complexes $RPerf$ is a derived Artin stack which admits a filtration by open sub stacks $RPerf^{[a,b]}$. The latter is a derived $(b-a+1)$-Artin stack. See:

Moduli of objects in dg-categories. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 40 (2007) no. 3, pp. 387-444. doi : 10.1016/j.ansens.2007.05.001. http://www.numdam.org/item/ASENS_2007_4_40_3_387_0/

This derived stack plays a critical role in the recently hot topic of shifted symplectic structures and shifted deformation quantization, see:

Pantev, T., Toën, B., Vaquié, M. et al. Publ.math.IHES (2013) 117: 271. https://doi.org/10.1007/s10240-013-0054-1