We study the relations between Multiplicative Exponential Linear Logic (
mELL) and Baillot-Mazza Linear Logic by Levels (
mL
3). We design a decoration-based translation between propositional
mELL and propositional
mL
3. The translation preserves the cut elimination. Moreover, we show that there is a proof net
P{\it \Pi} of second order
mELL that cannot have a representative
P¢{\it \Pi'} in second order
mL
3 under any decoration. This suggests that levels can be an analytical tool in understanding the complexity of second order
quantifier.