How Low can Approximate Degree and Quantum Query Complexity be for Total Boolean Functions?

It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Ω(log n), and that this bound is achieved for some functions. In this paper, we study the case of approximate degree and bounded-error quantum query complexity. We show that for these measures, the correct lower bound is Ω(log n/ log log n), and we exhibit quantum algorithms for two functions where this bound is achieved.