A Short List of Equalities Induces Large Sign-Rank

We exhibit a natural function $F_n$ on $n$ variables that can be computed by just linear-size decision list of “Equalities,” but whose sign-rank is $2^{\Omega(n^{1/4})}$. This yields the following two new unconditional complexity class separations. 1. Boolean circuit complexity. The depth-two threshold formulas when weights gates are unrestricted (${THR} \circ {THR}$), any ${THR} {MAJ}$ (the bottom polynomially bounded in $n$) computing requires size provides first separation between classes and {THR}$. While Amano Maruoka [Proceedings 30th International Symposium Mathematical Foundations Computer Science, 2005, pp. 107--118] Hansen Podolskii 25th Annual IEEE Conference Computational Complexity, 2010, 270--279] emphasized superpolynomial separations remained basic open problem, our fact exponential. In contrast, Goldmann, H\aastad, Razborov [Comput. 2 (1992), 277--300] showed more than twenty-five years ago functions efficiently computable ${MAJ} {THR}$ circuits also circuits. view this, it was not even clear if significantly powerful until work, there no candidate identified for potential separation. 2. Communication (under partition inputs) lies communication ${P}^{{MA}}$. Since has large sign-rank, this implies ${P}^{{MA}} \nsubseteq {UPP}$, strongly resolving recent problem posed Göös, Pitassi, Watson 27 (2018), 245--304]. order to prove main result, we as an ${XOR}$ develop technique lower bound such functions. novel approximation-theoretic arguments against polynomials degree. Further, work highlights time “decision lists exact thresholds” common frontier making progress longstanding problems