APN functions provide optimal resistance to differential attacks when used as building blocks in block ciphers and are thus of great theoretical interest. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN functionFoperating onn= 2kbits is uniquely associated with a vector space partition of Fn2and a specific blocking set in the corresponding projective space PG(n−1, 2). These connections allow us to prove a variety of results on the Walsh spectrum ofF. We prove for instance thatFcan have at most one component function of amplitude larger than 23n/4. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and provide conditions for a function to be CCZ-equivalent to a permutation based on its number of bent components.