Q143. "What are the duties of the Chief Justice of the Supreme Court?"
A. The Chief Justice of the Supreme Court is only mentioned once in the Constitution. The mention indicates that the Framers assumed there would be a Chief, and that the Chief would have more to do than the one thing; but beyond the one Constitutional duty, the Constitution is silent about the Chief. The one thing, by the way, is buried in Article 1, Section 3: the Chief Justice presides over any impeachment trial involving the President.
That having been said, the Chief Justice does have other duties, but they are by law, rule, or tradition, and not a part of the Constitution. In the Supreme Court, the Chief has one vote, just like any other justice. If the Chief is in the majority on any vote, he decides who writes the opinion of the majority (this duty falls to the most senior justice of the majority if the Chief is in the minority). The Chief begins questioning in oral arguments and sits in the center of the array of justices in Court. He also is the "chair" of the conferences the justices have in closed session. The Chief is the person who traditionally swears in presidents at inaugural ceremonies. The Chief is also the head of the federal judiciary, and is often at the phalanx calling for things like raises for federal judges and more security in federal courtrooms.