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We establish a precise relationship between spherical harmonics and Fourierbasis functions over a hypercube randomly embedded in the sphere. Inparticular, we give a bound on the expected Boolean noise sensitivity of arandomly rotated function in terms of its "spherical sensitivity," which wedefine according to its evolution under the spherical heat equation. As anapplication, we prove an average case of the Gotsman-Linial conjecture,bounding the sensitivity of polynomial threshold functions subjected to arandom rotation.

Heat and Noise on Cubes and Spheres: The Sensitivity of Randomly Rotated Polynomial Threshold Functions