The low-temperature structural evolution and dielectric and dynamic properties of perovskite La1∕2Na1∕2TiO3 (LNTO) have been investigated by the means of x-ray and neutron diffractions, dielectric measurement, and first-principles density-functional calculations. The structural evolution has been clarified and reveals that LNTO behaves as a high-temperature quantum paraelectric, with a structure isomorphous to that of LaTiO3 (space group Pnma). Using ab initio methods, two chemical orderings, respectively, along the [001] and [111] directions of the perovskite structure, have been studied, first in their corresponding high-symmetry structures and then in low-symmetry tilted structures, analogous to I4∕mcm and Pnma. For each case, the optic dielectric tensor, the Born effective charges, and the phonon modes at various points of the Brillouin zone are computed, revealing several instabilities with different natures. In the low-symmetry phases studied, weak ferroelectric instabilities suggest that ordered LNTO structures are quantum paraelectrics. The chemical disorder on the A site is evidenced to decrease the dielectric response and the structural distortions. The calculations are compared to the experimental situation and are discussed in relation to other quantum paraelectrics.