Byzantine Agreement (BA) is one of the most fundamental problems in distributed computing, and its communication complexity is an important efficiency metric. It is well known that quadratic communication is necessary for BA in the worst case due to a lower bound by Dolev and Reischuk. This lower bound has been shown to be tight for the unauthenticated setting with $f < n/3$ by Berman et al. but a considerable gap remains for the authenticated setting with $n/3 \le f < n/2$. This paper provides two results towards closing this gap. Both protocols have a quadratic communication complexity and have different trade-offs in resilience and assumptions. The first protocol achieves the optimal resilience of $f < n/2$ but requires a trusted setup for threshold signature. The second protocol achieves near optimal resilience $f \le (1/2 - \varepsilon)n$ in the standard PKI model.