Let F⊆2[n] be a intersecting Sperner family (i.e. A¬⊂B, A∩B ≠∅ for all A,B∈F) with profile vector (fi)i=0…n (i.e. fi=|F∩binom([n], i)|). We present quadratic inequalities in the fi's which sharpen the previously known linear LYM-type inequalities.