Single-photon cameras present a promising avenue for high-resolution 3D imaging. They have ultra-high sensitivity---down to individual photons---and can record photon arrival times with extremely high (sub-nanosecond) resolution. Single-photon 3D cameras estimate the round-trip time of a laser pulse by forming equi-width (EW) histograms of detected photon timestamps. Acquiring and transferring such EW histograms requires high bandwidth and in-pixel memory, making SPCs less attractive for 3D-perception applications in resource-constrained settings such as mobile devices and AR/VR headsets. In this work we propose a new 3D sensing technique based on equi-depth (ED) histograms. ED histograms compress timestamp data more efficiently than EW histograms, reducing the bandwidth requirement. Moreover, to reduce the in-pixel memory requirement, we propose a lightweight algorithm to estimate ED histograms in an online fashion without explicitly storing the photon timestamps. This algorithm is amenable to future in-pixel implementations. We propose algorithms that process ED histograms to perform 3D computer-vision tasks of estimating scene distance maps and performing visual odometry under challenging conditions such as high ambient light. Our work paves the way towards lower bandwidth and reduced in-pixel memory requirements for SPCs, making them attractive for resource-constrained 3D vision applications.