I am in SoCal. Same issue here on both my verizon iPhone 5 phones, first when I purchased them and now its back, just had the phones replaced again tonight. One definite but lame workaround is to turn of LTE.
Upon initial purchase of white iPhone, consistent/nightly time issues where the time was off by hours, standby time showed millions of hours and the date was reset to Sept or August, causing missed morning alarms, alarms in general to go off expectantly, calendar events from the past to pop up, SMS messages would not send due to incorrect date/time, &emails sent with invalid time stamps . Turned off LTE and the issue was no longer present, however having LTE is important to me. I spent multiple hours on the phone with Verizon & Apple as well as multiple trips to the nearest Apple store, 20 miles away. Had the phone replaced on 2 separate occasions, even asked for a black one to give me a better feel that it may have different vendor chips inside or just out of luck it was only with the white iPhones. On the last replacement(which both Apple and Verizon refused to provide me a black phone), the issue was no longer present for the week prior to the 6.0.1 update. As soon as the update came out, I performed the update and still did not have any issues and thought that the whole thing was taken care of.
I then purchased a black iPhone 5 and performed the 6.0.1 update. No issues since the latest replacement, purchase of the black iPhone, & the 6.0.1 update.
Then beginning on the morning of 12/17 my wife awoke in a panic as she was late for work and the phone alarm had not gone off since it showed 5:50 am and it was really 7am. I noticed the same on my phone. Since then, every morning, the time is off by approximately 1hr behind. Last night, after being out of town all day in another city 20 miles away and prior to driving home, I noticed my time was off by 1 hr or so. I cycled cellular data off and then back on, as well as airplane mode on then off and the time updated on the phone.
To throw this out there as well, I personally have multiple devices pre iPhone 5with only 3G and I do not and have not had the issue. It happens mostly at night, however also has happened a few time during the day in other cities.
I am in touch with Verizon Engineering and Apple Engineering via E-mail as this happens daily.
Last chance Apple\Verizon or I want a refund and go to a Galaxy.