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All the measurements were performed at sample temperatures of 10 K in helium flow cryostats. The resonant excitation was done with a titanium sapphire femtosecond laser featuring a bandwidth of 100 fs and a repetition rate of 80 MHz passing a 4f pulse-shaper setup to form picosecond pulses (10 ps). The laser was focused via an objective (numerical aperture of 0.42) onto the sample. We used tunable notch filters in transmission with a bandwidth of 0.4 nm to reject scattered laser light. Before fiber coupling of the QD signal, the light was guided to and spectrally filtered by another set of tunable notch filters used in reflection to separate X and XX emission. The signal was then detected either by a charge-coupled device camera or—during photon correlation spectroscopy—by APDs connected to the correlation electronics. The temporal resolution of these detectors is about 500 ps (40 dark counts per second). Combinations of λ/2, λ/4 and fixed polarizers were properly placed to estimate the entanglement or teleportation fidelity. The two-photon interference experiments rely on a single-mode fiber BS with reflectance R = 49.0(1)%, transmittance T = 51.0(1)%, and mode overlap 1 - ε = 96(1)%. The measurements of the FSS were carried out with a rotating λ/2 waveplate in front of a fixed polarizer placed at the entrance of a grating spectrometer (1800 lines/mm). Using Gaussian fitting of both the X and XX lines, the FSS can be measured with sub-micro-electronvolt resolution. Lifetime measurements were performed using another APD featuring a temporal resolution of 50 ps.

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