We exhibit a natural function F, that can be computed by just a linear sized decision list
of'Equalities', but whose sign rank is exponentially large. This yields the following two new
unconditional complexity class separations. The first is an exponential separation between
the depth-two threshold circuit classes Threshold-of Majority and Threshold-of-Threshold,
answering an open question posed by Amano and Maruoka [MFCS'05] and Hansen and
Podolskii [CCC'10]. The second separation shows that the communication complexity class …

We exhibit a natural function F_n on n variables that can be computed by just a linear-size
decision list of “Equalities,” but whose sign-rank is 2^Ω(n^1/4). This yields the following two
new unconditional complexity class separations. 1. Boolean circuit complexity. The function
F_n can be computed by linear-size depth-two threshold formulas when the weights of the
threshold gates are unrestricted (THR∘THR), but any THR∘MAJ circuit (the weights of the
bottom threshold gates are polynomially bounded in n) computing F_n requires size …