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The Hall-magnetohydrodynamic (Hall-MHD) system plays a significant role in the fluid description of an astrophysical plasma that features fast magnetic reconnection. Compared to the classical MHD system, the appearance of Hall term brings much more challenges in the field of numerical treatment. The aim of this paper is to construct an efficient spectral scheme for incompressible Hall-MHD system on three-dimensional infinite domain R 3. Combining with a novel (H N + 2 , H N) -Hermite-Galerkin spectral scheme for spatial approximation, a new non-zero constant function approach for the nonlinear terms, a second-order projection method for the momentum equations, and BDF2 scheme for the decoupled system, here we establish, for the first time, a fully decoupled, completely linearized, and unconditionally energy stable scheme with directly spatial approximation in R 3 and second-order temporal accuracy for incompressible Hall-MHD system. Numerical results are presented to illustrate the features of the proposed scheme, including convergence/stability tests and simulations of O- and X-points appearing in fast magnetic reconnection within the Hall-MHD regime. • We develop a novel (H N + 2 , H N)-Hermite-Galerkin spectral scheme for Hall-MHD system. • The scheme is linear, second-order time accurate, and unconditionally energy stable. • A new ordinary differential equation is established to reduce computational cost. • All the unknown quantities are decoupled in the practical computation. • We reveal new features of second-order pressure-projection method on infinite domain. [ABSTRACT FROM AUTHOR]