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A first goal of this paper is to precisely relate the homotopy theories ofbialgebras and $E_2$-algebras. For this, we construct a conservative and fullyfaithful $\infty$-functor from pointed conilpotent homotopy bialgebras toaugmented $E_2$-algebras which consists in an appropriate "cobar" construction.Then we prove that the (derived) formal moduli problem of homotopy bialgebrasstructures on a bialgebra is equivalent to the (derived) formal moduli problemof $E_2$-algebra structures on this "cobar" construction. We show consequentlythat the $E_3$-algebra structure on the higher Hochschild complex of this cobarconstruction, given by the solution to the higher Deligne conjecture, controlsthe deformation theory of this bialgebra. This implies the existence of an$E_3$-structure on the deformation complex of a dg bialgebra, solving along-standing conjecture of Gerstenhaber-Schack. On this basis we solve along-standing conjecture of Kontsevich, by proving the $E_3$-formality of thedeformation complex of the symmetric bialgebra. This provides as a corollary anew proof of Etingof-Kazdhan deformation quantization of Lie bialgebras whichextends to homotopy dg Lie bialgebras and is independent from the choice of anassociator. Along the way, we establish new general results of independentinterest about the deformation theory of algebraic structures, which shed a newlight on various deformation complexes and cohomology theories studied in theliterature.

Deformation theory of bialgebras, higher Hochschild cohomology and formality