It comes as Laura Trott also quit as a ministerial aide, saying "trust in politics is - and must always be - of the utmost importance, but sadly in recent months this has been lost".
Meanwhile schools minister Robin Walker has resigned, saying the Government has been "overshadowed by mistakes and questions about integrity".
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Hide AdMr Quince’s letter said: "Dear Prime Minister. Thank you for meeting with me and for your sincere apology regarding the briefings I received from No 10 ahead of Monday's media round, which we now know to be inaccurate.
"It is with great sadness and regret that I have no choice but to tender my resignation as minister for children and families as I accepted and repeated those assurances in good faith.
"It has been an honour to serve in government at both the Department for Work and Pensions and the Department for Education.
"Reaching this decision has not been easy. Stepping away from a job I love, where we are working every day to improve the life chances of vulnerable and disadvantaged children and young people up and down our country, pains me greatly.
"I will miss it hugely but pledge to do all I can to continue this important work from the backbenches.
"I would like to take this opportunity to put on record my sincere thanks to the hundreds of dedicated and hard-working civil servants with whom it has been a pleasure to work."