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Hard sphere decoding (HSD) has well-appreciated merits for near-optimal demodulation of multiuser, block single-antenna or multi-antenna transmissions over multi-input multi-output (MIMO) channels. At increased complexity, a soft version of sphere decoding (SD), so-termed list SD (LSD), has been recently applied to coded layered space-time (LST) systems enabling them to approach the capacity of MIMO channels. By introducing a novel bit-level multi-stream coded LST transmitter along with a soft-to-hard conversion at the decoder, we show how to achieve the near-capacity performance of LSD, and even outperform it as the size of the block to be decoded (M) increases. Specifically, for binary real LST codes, we develop exact max-log-based SD schemes with M + 1 HSD steps, and an approximate alternative with only one HSD step to trade off performance for average complexity. These schemes apply directly to the real and imaginary parts of quaternary phase-shift keying signaling, and also to quadrature amplitude modulation signaling after incorporating an appropriate interference estimation and cancellation module. We corroborate our near-optimal soft detection (SoD) algorithms based on HSD (SoD-HSD) with simulations.