  We consider scattering of massless higher-spin particles in the eikonal
regime in four dimensions. By demanding the absence of asymptotic
superluminality, corresponding to positivity of the eikonal phase, we place
constraints on the possible cubic couplings which can appear in the theory. The
cubic couplings come in two types: lower-derivative non-abelian vertices, and
higher-derivative abelian vertices made out of gauge-invariant curvature
tensors. We find that the abelian couplings between massless higher spins lead
to an asymptotic time advance for certain choices of polarizations, indicating
that these couplings should be absent unless new states come in at the scale
suppressing the derivatives in these couplings. A subset of non-abelian cubic
couplings are consistent with eikonal positivity, but are ruled out by
consistency of the four-particle amplitude away from the eikonal limit. The
eikonal constraints are therefore complementary to the four-particle test,
ruling out even trivial cubic curvature couplings in any theory with a finite
number of massless higher spins and no new physics at the scale suppressing
derivatives in these vertices.
