Near Optimal Alphabet-Soundness Tradeoff PCPs

We show that for all $\varepsilon>0$, for sufficiently large prime power $q$, for all $\delta>0$, it is NP-hard to distinguish whether a 2-Prover-1-Round projection game with alphabet size $q$ has value at least $1-\delta$, or value at most $1/q^{(1-\epsilon)}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for 2-query PCPs with alphabet size $q$, improving upon a result of [Chan 2016]. Our result has the following implications: 1) Near optimal hardness for Quadratic Programming: it is NP-hard to approximate the value of a given Boolean Quadratic Program within factor $(\log n)^{(1 - o(1))}$ under quasi-polynomial time reductions. This result improves a result of [Khot-Safra 2013] and nearly matches the performance of the best known approximation algorithm [Megrestki 2001, Nemirovski-Roos-Terlaky 1999 Charikar-Wirth 2004] that achieves a factor of $O(\log n)$. 2) Bounded degree 2-CSP's: under randomized reductions, for sufficiently large $d>0$, it is NP-hard to approximate the value of 2-CSPs in which each variable appears in at most d constraints within factor $(1-o(1))d/2$ improving upon a recent result of [Lee-Manurangsi 2023]. 3) Improved hardness results for connectivity problems: using results of [Laekhanukit 2014] and [Manurangsi 2019], we deduce improved hardness results for the Rooted $k$-Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity $k$-Route Cut Problem.