We develop the deformation theory of cohomological field theories (CohFTs),
which is done as a special case of a general deformation theory of morphisms of
modular operads. This leads us to introduce two new natural extensions of the
notion of a CohFT: homotopical (necessary to structure chain-level
Gromov–Witten invariants) and quantum (with examples found in the works of
Buryak–Rossi on integrable systems). We introduce a new version of
Kontsevich’s graph complex, enriched with tautological classes on the moduli
spaces of stable curves. We use it to study a new universal deformation group
which acts naturally on the moduli spaces of quantum homotopy CohFTs, by
methods due to Merkulov–Willwacher. This group is shown to contain both the
prounipotent Grothendieck–Teichm\”uller group and the Givental group.

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