Arkadev Chattopadhyay, Nikhil Mande

We show a new duality between the polynomial margin complexity of $f$ and the discrepancy of the function $f \circ$ XOR, called an XOR function. Using this duality,

we develop polynomial based techniques for understanding the bounded error (BPP) and the weakly-unbounded error (PP) communication complexities of XOR functions. ...
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Mark Bun, Robin Kothari, Justin Thaler

The approximate degree of a Boolean function $f$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. The approximate degree of $f$ is known to be a lower bound on the quantum query complexity of $f$ (Beals et al., FOCS 1998 and ... more >>>