Schaefer is correct. The US uses apostille in verying documents in each state when they are needed outside the US. China may not be a part of it, but that doesn't mean the China consulate doesn't want that state seal on it. Quite the opposite it seems, as it's part of their internal procedures.
Yes, I assume that Chinese consulates in USA recognize the state apostilles.
In other countries you also need the stamp from the country's Ministry of Foreign Affairs ( = Secretary of State in USA), but it's not called apostille.
Apostille in that case would by (Hague convention) definition mean that the (Chinese) consulate step could be skipped altogether.
So in these countries it is called simply "legalization" or "authentication" of the document.
For NZ specifically, please visit this website:
According to the website, both the NZ government and the foreign consulates in NZ accept "self-addressed envelope or pre-paid courier pack" - in some countries this requires visit in person (by you or agent).
As of 2017, China and the USA do NOT have Apostile agreements - so they use the somewhat more onerous authentication process.
Faking documents is easy - so I can understand the desire to ensure core work documents are somewhat vetted.