This month, we have had a few reports of problems logging in to Blogger, possibly involving the "cookies" which retain login information from one online session to the next.
Many security issues are deeply frustrating, to the folks who have to endure the problems. This becomes more frustrating, when you understand that the code from Blogger.Com has no choice in the matter. If Blogger.Com requires cookies (as most web sites do), and if cookies aren't being retained by the browser, the Blogger scripts can't function. And when you have a setting that's created under "Blogger.com" and read while your blog is being accessed (under "BlogSpot.com" or whatever non BlogSpot domain your blog is published), you'll need third party cookies enabled.
The really frustrating issue is that many security products, constantly "evolving" thanks to economic pressure (free enterprise competition) and security pressure (never ending hacking attempts), are constantly adding to their functionality. What used to be called a "personal firewall" (a network level protection) is now called a "security suite". Ditto for what used to be called "antivirus protection" (an application level protection) is likewise called a "security suite". Any of these may now contain cookie or script restrictions, and cause your problem.
Complicating this matter is some security suites may bind with some browsers, but not all, and may not consistently provide the same level of protection with all browsers.
If you are seeing this problem with one browser, but not another, I seriously suggest that you examine your computer for any security protection, and for any recent changes. Only you know what security products have been added to your computer. And any of them might have settings to block (or permit) cookies and scripts.
The bottom line, though, is that Blogger scripts don't manage the cookies, they only read and write them. If you have a problem, you visit Blogger Help Group, and you find only a couple (or no) other posts about your problem, consider carefully whether Blogger is undeniably the primary cause of your problem.