[jubjubbird]

The MEMS accelerometers in our phones are in theory sensitive to gravity changes, but they're not nearly sensitive enough. This group is trying to improve the MEMS sensing technology, but it's doubtful it will be as good as classical gravity meters (falling mass, or spring) or quantum (BEC) ones:

https://www.gla.ac.uk/research/beacons/nanoquantum/wee-gglas...

One important distinction is between measuring gravity (generally, acceleration in the vertical direction) and the gravity gradient. The latter is generally easier and more useful for navigation.