What is the mathematical meaning of locality?

"The special theory of relativity limits the speed at which all such influences can travel to the speed of light, c. Therefore, the principle of locality implies that an event at one point cannot cause a simultaneous result at another point."

But this has no "use" for me in scattering amplitude physics, so I looked more into it and, in a question on this website, I found this talk. What I understood from it was that locality implies conservation of momentum:

$$\sum p_i =0$$

Is this correct?

If so is it all I need to know about the concept of locality for scattering amplitudes, or is there another mathematical meaning behind it?

The locality condition in quantum field theory is that the commutator of the interaction density operator vanishes at space-like separations. For fermions, this means that field operators must appear in pairs, and the anticommutator $$\{\psi(x),\bar{\psi}(y)\}$$ vanishes at space-like separations.
Probably the main difference between this and locality in classical special relativity is that $$x$$ and $$y$$ do not refer to specific positions in a reference frame, but are subject to the principle of superposition, and integrated over, as in quantum theory generally.