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How is that better than just showing the flyer at the door to get the discount?

They can track response rates for particular flyers by pointing them at different URLs. Measurable A/B testing for physical advertising.

They can print A/B versions of flyers and collect them upon entry. No QR needed for this kind of measuring. :)

because you're likely to lose the flyer, and it's a PITA to carry it around. You're always gonna keep your mobile though.