Two substantial technological advances have reshaped the public square in recent decades: first with the advent of the internet and second with the recent introduction of large language models (LLMs). LLMs offer opportunities for a paradigm shift towards more decentralized, participatory online spaces that can be used to facilitate deliberative dialogues at scale, but also create risks of exacerbating societal schisms. Here, we explore four applications of LLMs to improve digital public squares: collective dialogue systems, bridging systems, community moderation, and proof-of-humanity systems. Building on the input from over 70 civil society experts and technologists, we argue that LLMs both afford promising opportunities to shift the paradigm for conversations at scale and pose distinct risks for digital public squares. We lay out an agenda for future research and investments in AI that will strengthen digital public squares and safeguard against potential misuses of AI.