<p>The conservation of energy and transverse momentum in the process of spontaneous parametric downconversion enforces on the generated entangled signal and idler photons strict inter-photon quantum correlations—with the spatial and temporal degrees of freedom separately exhibiting such correlations. Nonlinear crystals of realistic lengths enforce only loose intra-photon spatiotemporal correlations, which obscure the observation of any angular-dispersion-induced effects and have barred quantum optics from appropriating the recent progress in spatiotemporally structured light. Here we show that placing a spatiotemporal spectral modulator in the path of a heralded signal photon yields a single-photon ‘space-time wave packet’ that is endowed with a precisely structured spatiotemporal spectrum required for diffraction-free propagation at a tunable group velocity. Furthermore, by projecting the multimoded, spatiotemporally structured signal photon onto a single spatial mode to herald the arrival of the idler photon, we remotely prepare a diffraction-free single-photon space-time wave packet in the idler that did not traverse the spectral modulator. The signal/idler photons have complementary subluminal/superluminal group velocities as a consequence of their intrinsic inter-photon spectral anti-correlation, which reverses the sign of the spatiotemporal spectral curvature for the remotely prepared photon. These results pave the way for the applications of spatiotemporally structured light to impact quantum optics in the context of quantum communications, imaging and sensing.</p>