A study of the asymptotic decay of the pair radial correlations that can be defined in the path-integral quantum hard-sphere fluid is presented. These distinct quantum pair correlations arise from the breaking of the classical spherical symmetry of the particles under the quantum effects. The three types of correlations analyzed are the so-called linear response, instantaneous and center-of-mass, which correspond to distinct averaging criteria over the thermal packets associated with the quantum particles. The basic methodology employed to perform this analysis, based on the fixing of the complex poles of the static structure factor, was put forward by Tago and Smith [Can. J. Phys. 55, 761 (1977)] and independently by Evans et al. [Mol. Phys. 80, 755 (1993); J. Chem. Phys. 100, 591 (1994)]. To apply this method it is required the knowledge of the direct correlation functions connected to the pair radial correlations involved, which over a wide range of conditions are available in the literature [J. Chem. Phys. 108, 9086 (1998); Mol. Phys. 99, 585 (2001)]. In the quantum hard-sphere fluid both pure imaginary and complex conjugate poles are possible, and the properties of this system depend on the density and the temperature. However, no Fisher-Widom line has been obtained. The decay of the correlations in this fluid is of the exponentially damped oscillatory type, in agreement with the purely repulsive character of the interparticle potential. The linear response and instantaneous decay properties follow the same pattern, albeit slight differences can be observed. Comparison with the Percus-Yevick classical results and with those that can be derived from Tarazona and Vicente's-model [Mol. Phys. 56, 557 (1985)] is made. Besides, it is proven that the above theory of asymptotic behavior evinces the capability for resolving fine-drawn features of quantum changes of phase in the hard-sphere system.