  In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "${\cal F}$-consistent nonlinear expectation") to the case
when it is allowed to be dominated by a $g$-expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense, including the
Doob-Meyer type decomposition theorem and the optional sampling theorem. More
importantly, we show that any quadratic ${\cal F}$-consistent nonlinear
expectation with a certain domination property must be a quadratic
$g$-expectation. The main contribution of this paper is the finding of the
domination condition to replace the one used in all the previous works, which
is no longer valid in the quadratic case. We also show that the representation
generator must be deterministic, continuous, and actually must be of the simple
form.
