  A complex projective $t$-design is a configuration of vectors which is
``evenly distributed'' on a sphere in the sense that sampling uniformly from it
reproduces the moments of Haar measure up to order $2t$. We show that the set
of all $n$-qubit stabilizer states forms a complex projective $3$-design in
dimension $2^n$. Stabilizer states had previously only been known to constitute
$2$-designs. The main technical ingredient is a general recursion formula for
the so-called frame potential of stabilizer states. To establish it, we need to
compute the number of stabilizer states with pre-described inner product with
respect to a reference state. This, in turn, reduces to a counting problem in
discrete symplectic vector spaces for which we find a simple formula. We sketch
applications in quantum information and signal analysis.
