  Recent breakthroughs in quantum query complexity have shown that any formula
of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum
queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In
particular, this gives an upper bound on the approximate polynomial degree of
formulas of the same magnitude, as approximate polynomial degree is a lower
bound on quantum query complexity [BBCMW01].
  These results essentially answer in the affirmative a conjecture of O'Donnell
and Servedio [O'DS03] that the sign degree--the minimal degree of a polynomial
that agrees in sign with a function on the Boolean cube--of every formula of
size n is O(sqrt(n)).
  In this note, we show that sign degree is super-multiplicative under function
composition. Combining this result with the above mentioned upper bounds on the
quantum query complexity of formulas allows the removal of logarithmic factors
to show that the sign degree of every size n formula is at most sqrt(n).
