We prove that the P NP -type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the P NP -type communication complexity of a lifted version of f . As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture P NP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).