There is the option to "turn off location history" but what that does is prevent you from being able to see your timeline of locations. Google does, apparently, keep this data unless you hunt down a hidden option which is entirely ambiguous.
If you're interested there's a Wired article which explains exactly how to stop Google from keeping your location data.
But, this might be unnecessary now because 9to5Google has been digging through beta code in the new Google Maps update and has discovered that the company is testing a new mode called "automatically delete location history".
The mode will automatically delete your location history without the need for you to enter a pesky menu and do it yourself. This is a good but arguably long-overdue feature which gives users back some control over how Google manages private data.
The beta version is 10.10 and, of course, there's no guarantee that the mode will make it into any final version of the app. However Google must be worried that the EU or other nations regulators are going to start asking difficult and expensive questions.
With strict requirements on getting users to opt-in, rather than opt-out, Google could be risking expensive rulings against it that could see huge percentages of its global turnover taken from it in fines.
Besides, Google must, at this point, be aware that the public is becoming less and less tolerant to this kind of monitoring of their activities. It is presumably also aware that the people who get upset about this sort of thing will likely use a setting to stop it, but a larger group of people simply won't bother. It's the same reason Facebook doesn't yet encrypt Messenger - the public simply isn't demanding it vocally enough.
The Beta is live in the Google Play store although the options to start using this feature currently aren't.
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