Agreement protocols for partially synchronous or asynchronous networks tolerate fewer
than one-third Byzantine faults. If parties are equipped with trusted hardware that prevents equivocation, then fault tolerance can be improved to fewer than one-half Byzantine faults, but typically at
the cost of increased communication complexity. In this work, we present results that use small trusted
hardware without worsening communication complexity assuming the adversary controls a fraction of
the network that is less than one-half. Our results include a version of HotStuff that retains linear
communication complexity in each view and a version of the VABA protocol with quadratic communication, both leveraging trusted hardware to tolerate a minority of corruptions. Our results use expander
graphs to achieve efficient communication in a manner that may be of independent interest.