We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Letfbe anm-bit Boolean function and consider ann-bit functionFobtained by applyingfto conjunctions of possibly overlapping subsets ofnvariables. Iffhas quantum query complexityQ(f), we give an algorithm for evaluatingFusingO~(Q(f)⋅n)quantum queries. This improves on the bound ofO(Q(f)⋅n)that follows by treating each conjunction independently, and our bound is tight for worst-case choices off. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree off.By recursively applying our composition theorems, we obtain a nearly optimalO~(n1−2−d)upper bound on the quantum query complexity and approximate degree of linear-size depth-dAC0circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0circuits.As an additional consequence, we show that AC0∘⊕circuits of depthd+1require sizeΩ~(n1/(1−2−d))≥ω(n1+2−d)to compute the Inner Product function even on average. The previous best size lower bound wasΩ(n1+4−(d+1))and only held in the worst case (Cheraghchi et al., JCSS 2018).