We show that quasi-long-range (QLR) pair correlations that decay asymptotically with scaling r(-(d+1)) in d-dimensional Euclidean space R(d), trademarks of certain quantum systems and cosmological structures, are a universal signature of maximally random jammed (MRJ) hard-particle packings. We introduce a novel hyperuniformity descriptor in MRJ packings by studying local-volume-fraction fluctuations and show that infinite-wavelength fluctuations vanish even for packings with size and shape distributions. Special void statistics induce hyperuniformity and QLR pair correlations.