None identify buddha nature with the past. Generally speaking and regardless of the school of thought, buddha nature points to you having (at present) the potential to achieve buddhahood.
Cittamatrin identify buddha-nature to the mind-basis-of-all (alaya vijnana) while Madhyamika identify it to suchness with defilements, that is the emptiness of the mind of a sentient being. The Lankavatara Sutra equates tathagatagarba with mind-basis-of-all.
In his commentary to Maitreya's Sublime Continuum, Gyaltsab Je (Madhyamika-Prasangika) writes:
Suchness with defilement is the basic element not liberated from the
obscurations of mental afflictions, and is known as tathagata essence (Skt. tathagatagarbha).
Madhyamika-Svatantrika speak of emptiness of true existence while Madhyamika-Prasangika speak of emptiness of inherent existence. The Madhyamika idea is basically that if your mind was not empty, it would not be dependent, no change could occur in it, it could not be subject to transformation, it could not be separated from the obscuring factors, it could not come to be endowed with the two purities, would not be suitable to transform into the bodies of a buddha, and so forth.