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California has the nation’s strictest gun laws. There is a waiting period, a written test and a number of restrictions on guns and gear. Some voters pondering Proposition 63 have asked: Haven’t we done enough?

Gavin Newsom, the lieutenant governor, says no. He’s developed the initiative that would push the state’s gun limits even further.

Under Proposition 63, ammunition purchasers would be treated similar to gun purchasers. Before a Californian could buy even a single bullet of any kind, he or she would need to pay a fee of up to $50 and wait up to 30 days to obtain a four-year permit.

“The game changer in the country is going to be ammunition background checks,” Mr. Newsom, who is running for governor in 2018, said in a phone interview.