  We give a hierarchy of $k$-point bounds extending the
Delsarte-Goethals-Seidel linear programming $2$-point bound and the
Bachoc-Vallentin semidefinite programming $3$-point bound for spherical codes.
An optimized implementation of this hierarchy allows us to compute~$4$, $5$,
and $6$-point bounds for the maximum number of equiangular lines in Euclidean
space with a fixed common angle.
