Condo property taxes and abatements question

I am comparing the current tax bills on two very similar condos in Manhattan. One has it's taxable assessed value reduced due to code 1017-01 "SCHOOL TAX RELIEF" by about 7 percent. Then it has an abatement "Co-op Condo Abatement" of about 17 percent. The other condo has neither. As far as I can tell the assessment reduction is the STAR abatement and the Co-op condo abatement is for primary residents (with some other restrictions). Is this correct and would it mean that a new owner in either condo would be eligible for the same abatements and reductions? It looks like the primary resident Condo Co-op abatement takes about a year to go through (file by March for summer effect) - is there any way to get it sooner or retroactively?