What is heretical about the (Assyrian/Ancient) Church Of the East?
It's kind of like with the OO-EO split, the same kinds of terms were involved.
EOs, OOs, and the ACE/"Nestorians" agree that Christ has two essences, and is one person.
Chalcedon said that Christ is in and has two natures - human and divine ones - and only one hypostasis, which is human and divine.
The OOs historically rejected Christ as being in or having two natures, a denial that seen by EOs and Nestorians as implying a denial of his human nature, if not human essence.
The Nestorians taught that Christ has two "hypostases", a teaching that was seen by EOs and OOs as entailing Christ is two separate persons.
Currently the churches involved are in dialogue on the issue. Whether these churches are seen as heretical looks perhaps dependant on whether one focuses on the aspect of the debate that unites us (Christ being fully man and fully god in one person) or over which we have disagreed (the number of natures and hypostases). If one focuses on the fact that we all agree that Christ has one person, then one is less likely to see Nestorians as teaching Christ has two persons - the heretical implication of their teaching on hypostases when taken alone.
It's like someone promising to pick you up from the train station on Thursday the 2nd of the month, when Thursday is the 1st of the month. Your train will arrive on Thursday the 1st, so you like to expect that they will be there. At least that's what I found when talking with one of the more hardcore non-Chalcedonians on this question on another forum.