Alexander Levin was born in Tel Aviv on December 6, 1995, the youngest of the four sons of the playwright and director Hanoch Levin. His mother is the actress Lilian Barto, who met Hanoch Levin in 1993 during rehearsals for his play The Child Dreams. In July 1996 his father wrote a poem for the infant, "A Week in the Life of the Child Hagezer," using the private nickname he had given him; Hanoch Levin died in August 1999, when Alexander was three years and eight months old. He gave his first newspaper interview about that inheritance only in 2021, and framed it narrowly: the playwright belongs to everybody, the man was his father.

The other constant in his life started on the same day. The drummer Eviatar Slivnik was born on December 6, 1995 in the same Tel Aviv maternity ward, a coincidence their mothers discovered nine months later when the two boys were enrolled in the same daycare. From there the pair moved through the same institutions in sequence: the music track of a Tel Aviv arts school, the jazz track at the Thelma Yellin High School of the Arts, service as distinguished musicians in the Israel Defense Forces jazz ensemble, and then the Rimon School of Jazz and Contemporary Music, where Levin came through the excellence program known as the Jazz Institute. He collected twelve excellence scholarships along the way, including a full-tuition award to Berklee College of Music and an Israeli Conservatory scholarship in honor of Ran Baron.

His first recordings came at eighteen. In 2014 he was the tenor saxophonist of Jupiter, a young Tel Aviv sextet with trumpeter Ron Warburg, harmonica player Yotam Ben-Or, pianist Tomer Bar, bassist Alon Near and Slivnik on drums; the band contributed two tracks, "Shnatz" and "Jupiter," to All Original: Best Young Israeli Jazz, the compilation the bassist Avishai Cohen assembled for his Razdaz Recordz label and released on October 6, 2014. The same year he cut a guest chorus on the title track of pianist Ronen Shmueli's The Pike, a session that sat on the shelf until 2019.

In the summer of 2015 Omer Avital needed a saxophonist and heard Levin's name from several people. Levin was still in uniform. Avital called him in for a session and hired him, later describing what he heard as maturity and a deep understanding of the blues in a player barely out of his teens. Levin took an early discharge, obtained a United States visa and moved to New York in 2016, where he stayed for the rest of the decade. He held one of the two tenor chairs in Avital's band throughout its recorded life, sharing the front line with Asaf Yuria across three albums: Abutbul Music (Jazz Village, 2016), with Yonathan Avishai on piano and Ofri Nehemya on drums; Qantar, recorded at Brooklyn Recording on January 8 and 9, 2017 and issued by Jazz Family on September 7, 2018; and New York Paradox, cut at Wilson Live in Brooklyn on April 21 and 22, 2019 with the five of them playing in one room without headphones, separation or edits, and released by Jazz&People and Zamzama on February 28, 2020. The touring that went with it took him to the Marciac and Gent jazz festivals, Ronnie Scott's, New Morning and the Duc des Lombards, and to Dizzy's Club and the Apollo Theater in New York. One leg of it, a Barcelona date with Yuria followed by a French booking, collapsed when the airline refused to board their saxophones: the two rented a car, watched its engine burn out on a mountain between Spain and France in the middle of the night, waited three hours for a phone signal, and drove another fourteen hours to make the gig. Back in New York he was a regular at Smalls, where the club's archive lists him at Hillel Salem's after-hours jam sessions from June 2016 onward, at a Yuria after-hours set in June 2018 and with the Omer Avital Septet in August 2021. In Israel he also took the tenor chair on four of the nine tracks of the Israel Jazz Orchestra's Nine Tales One Story, cut at Bardo Studios in Ramat Gan and released on February 25, 2018.

The pandemic sent him home in March 2020. Rather than wait it out, he and Slivnik ran a drive-in jazz series on Saturday afternoons in a Tel Aviv parking lot, loading in every week, drawing sixty to seventy cars by sunset and rotating guests that included Eli Degibri, Shai Maestro and Yoni Rechter. The two mothers organized it. In June 2021 the pair closed the Israeli chapter with a set at the seventh Jerusalem Jazz Festival, held June 22 to 24 in the sculpture garden of the Israel Museum under the artistic direction of trumpeter Avishai Cohen, playing a quartet program of their own material with Tom Oren on piano and Yonatan Levi on bass. Levin returned to New York the following month, and in September 2021 recorded the tenor parts for pianist Hila Kulik's More Than a Change at Sear Sound.

Hand Made, his debut as a leader, came out on September 13, 2022. He wrote all nine pieces, produced it himself and cut it in Israel across two days at Pluto Studios with Eden Ladin on piano, Yonatan Levi on bass and Slivnik on drums, Yaron Mohar engineering and Dave Darlington mixing and mastering; Slivnik's own debut, recorded in the same stretch, has Levin on it in return. Three tracks went out as advance singles in the fortnight before the album: "Lexington Avenue" on August 30, "Tamar's Love Song" on September 6 and "Higher Than Myself" on September 8. Avishai Cohen then invited him to lead a set of his own music at the Jerusalem Jazz Festival. From 2022 Levin also turned toward film and television scoring, taking four film-score and orchestration courses at Berklee and UCLA and writing for short films, an RBCO fashion piece and the Israeli drama series Red Skies, which ran on Channel 13 from June to August 2023.

Levin is a straight-ahead tenor player whose reference points are American and mid-century rather than local. Slivnik, who has played with him since they were four, described the Hand Made sessions as the closest thing to New York that could be made in Israel and called the writing "completely black jazz," and Levin's own account of Avital's mentorship is about permission rather than vocabulary: he was tense and unhappy with his takes on his first record with the band until Avital told him the reason he had been hired was that he tries things. What Avital says he heard in the conscript he auditioned was blues feeling, which is the quality most reviewers have picked up on since. Reviewing New York Paradox for DownBeat in August 2020, Sean J. O'Connell heard the two tenors "pop with tight staccato support and unadorned blues," and Raul Da Gama, writing for Jazz da Gama, described Avital as employing Levin and Yuria to carry the lead voices in each melody and harmony. The Qantar book also pulled him well outside bebop syntax, into Moroccan and Yemenite material where the horns sit inside a groove instead of on top of it. As a composer he says he does not set texts and has never written lyrics, but hears hidden ones behind his own melodies.