It will be interesting to see if and how states will use testing post-lockdown.
I live in Germany, a country governed in a similar way with a federal government and state governments that are responsible for health directives. We encountered similar issues concerning the lack of overall country-wide consensus regarding lockdown regulations, but these were ironed out (plus Merkel put her foot down at one point and started taking back some controls).
The approach here was tests, tests and more tests. Test centres, mobile test units, tests from a man on a bike, you name it... Tests became mandatory in secondary school (3 times a week, supervised by the class teacher at the beginning of the day). You also needed to show a negative test to enter restaurants, non-essential shops and other non-essential public places. Large settings (such as swimming pools etc) still require either a vaccine certificate or negative test for anyone 12 and over.
The other strict mandate was, and still is here in Bavaria, though other states are loosening up, FFP2 masks in any indoor setting, including in secondary schools. Numbers have been low for a while now, but it was only in the past month that my DS remove his mask during lessons. They still have to wear them when moving around the school, and have been asked to wear them fully for the first 2 weeks of term in September to catch any post-holiday cases. Testing will also still continue.
This strategy seems to have worked. Germany has very low numbers compared to many of its neighbours, yet has open borders with the usual restrictions on entrants from high-risk countries.