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We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with $$ \mathcal{N}\ge 4 $$ supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In $$ \mathcal{N}=4 $$ SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite number of OPE coefficients in the interacting SCFT (U(2)2 × U(1)−2 ABJ theory) that constitutes the IR limit of O(3) $$ \mathcal{N}=8 $$ super-Yang-Mills theory. More generally, in $$ \mathcal{N}=8 $$ SCFTs with a unique stress tensor, we are led to conjecture that many superconformal multiplets allowed by group theory must actually be absent from the spectrum, and we test this conjecture in known $$ \mathcal{N}=8 $$ SCFTs using the superconformal index. For generic $$ \mathcal{N}=8 $$ SCFTs, we also improve on numerical bootstrap bounds on OPE coefficients of short and semi-short multiplets and discuss their relation to the exact relations between OPE coefficients we derived. In particular, we show that the kink previously observed in these bounds arises from the disappearance of a certain quarter-BPS multiplet, and that the location of the kink is likely tied to the existence of the U(2)2 × U(1)−2 AJ theory, which can be argued to not possess this multiplet.