  High-dimensional entanglement provides unique ways of transcending the
limitations of current approaches in quantum information processing, quantum
communications based on qubits. The generation of time-frequency qudit states
offer significantly increased quantum capacities while keeping the number of
photons constant, but pose significant challenges regarding the possible
measurements for certification of entanglement. Here, we develop a new scheme
and experimentally demonstrate the certification of 24-dimensional entanglement
and a 9-dimensional quantum steering. We then subject our photon-pairs to
dispersion conditions equivalent to the transmission through 600-km of fiber
and still certify 21-dimensional entanglement. Furthermore, we use a steering
inequality to prove 7-dimensional entanglement in a semi-device independent
manner, proving that large chromatic dispersion is not an obstacle in
distributing and certifying high-dimensional entanglement and quantum steering.
Our highly scalable scheme is based on commercial telecommunication optical
fiber components and recently developed low-jitter high-efficiency
single-photon detectors, thus opening new pathways towards advanced large-scale
quantum information processing and high-performance, noise-tolerant quantum
communications with time-energy measurements
