  Due to their strong resonances with their host planet, Trojan asteroids can
remain in stable orbits for billions of years. As a result, they are powerful
probes for constraining the dynamical and chemical history of the solar system.
Although we have detected thousands of Jupiter Trojans and dozens of Neptune
Trojans, there are currently no known long-term stable Earth Trojans. Dynamical
simulations show that the parameter space for stable Earth Trojans in
substantial, so their apparent absence poses a mystery. This work uses a large
ensemble of N-body simulations to explore how the Trojan population dynamically
responds if Earth suffers large collisions, such as those thought to have
occurred to form the Moon and/or to have given Earth its Late Veneer. We show
that such collisions can be highly disruptive to the primordial Trojan
population, and could have eliminated it altogether. More specifically, if
Earth acquired the final 1\% of its mass through ${\cal O}(10)$ collisions,
then only $\sim1\%$ of the previously bound Trojan population would remain.
