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Qantas if offer cheap flights from Queenstown to LA for $997. It's rare that we see deals from Queenstown but this seems to be the cheapest flight to LA from Queenstown seen on Cheapies by $16. Dates are March and May to October 2019 for SFO; August only for LAX. Flights connect through Australia.
San Francisco $1004 - Skyscanner - Google Flights
Los Angeles $1007 - Skyscanner - Google Flights