I refute the dream XOR lemma conjecture as stated in a 1995 paper by Goldreich, Nisan, and Wigderson. The counterexample applies even to an XOR lemma for low-degree polynomials. I give new, very simple proofs of the XOR lemma which improve on the parameters of previous proofs. For example, the loss in circuit size is reduced quadratically.

Added new proof with improved parameters. Merged with separate technical report "The dream xor lemma is false."

I give an alternative proof of the xor lemma which may provide a simple explanation of why xor-ing decreases correlation.