We study two conjectures posed in the analysis of Boolean functions $f : \{-1, 1\}^n ? \{?1, 1\}$, in both of which, the Majority function plays a central role: the "Majority is Least Stable" (Benjamini et al., 1999) and the "Non-Interactive Correlation Distillation for Erasures" (Yang, 2004; O'Donnell and Wright, ... more >>>