  We study various aspects of the M-theory uplift of the $A_{N-1}$ series of
$(2,0)$ CFTs in 6d, which describe the worldvolume theory of $N$ M5 branes in
flat space. We show how knowledge of OPE coefficients and scaling dimensions
for this CFT can be directly translated into features of the momentum expansion
of M-theory. In particular, we develop the expansion of the four-graviton
S-matrix in M-theory via the flat space limit of four-point Mellin amplitudes.
This includes correctly reproducing the known contribution of the $R^4$ term
from 6d CFT data. Central to the calculation are the OPE coefficients for
half-BPS operators not in the stress tensor multiplet, which we obtain for
finite $N$ via the previously conjectured relation [arXiv:1404.1079] between
the quantum ${\cal W}_N$ algebra and the $A_{N-1}$ $(2,0)$ CFT. We further
explain how the $1/N$ expansion of ${\cal W}_N$ structure constants exhibits
the structure of protected vertices in the M-theory action. Conversely, our
results provide strong evidence for the chiral algebra conjecture.
