Recently, Quastel and Remenik (2019 (arXiv:1908.10353)) found a remarkable relation between some solutions of the finite time Kardar–Parisi–Zhang (KPZ) equation and the Kadomtsev–Petviashvili (KP) equation. Using this relation we obtain the large deviations at large time and at short time for the KPZ equation with droplet initial conditions, and at a short time with half-Brownian initial conditions. It is consistent with previous results and allows us to obtain sub-leading corrections, as well as results at intermediate time. In addition, we find that the appropriate generating function associated to the full Brownian initial condition also satisfies the KP equation. Finally, generating functions for some linear statistics of the Airy point process are also found to satisfy the KP property, and consequences are discussed.