Phone Scoop says Sprint’s iPhone 5 (not to be confused with the versions for AT&T, Verizon or the rest of the world) doesn’t support “Sprint’s version” of HD voice. However, VoLTE should be an option if HD Voice News is interpreting the tech specs correctly.
Sprint is moving to offer HD voice over an in-progress-upgraded 3G CDMA 1X Advanced network using Qualcomm’s EVRC-NW codec. (Which is looking more like another “Island of the Misfit Telecommunication Toys” issue if there’s no iPhone 5 support).
Assuming Sprint is offering Apple iPhone 5 CDMA model A1429*: the phone supports both CDMA and LTE, specifically Band 25 “owned” by Sprint. So this model would be capable of HD voice when Sprint rolled out Voice over LTE (VoLTE) in 2013.